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,f H A S W E L L'S 

ENGINEERS' AND MECHANICS' POCKET-BOOK, 

CONTAINING 

UNITED STATES AND FOREIGN WEIGHTS AND MEASURES ; 
TABLES OF AREAS AND CIRCUMFERENCES OF CIRCLES, CIRCULAR SEG- 
MENTS, AND ZONES OF A CIRCLE ; 
SQUARES AND CUBES, SQUARE AND CUBE ROOTS ; 
LENGTHS OF CIRCULAR AND SEMI-ELLIPTIC ARCS ; 
AND RULES OF ARITHMETIC. 

MENSURATION OF SURFACES AND SOLIDS; 

THE MECHANICAL POWERS ; 
GEOMETRY, TRIGONOMETRY, GRAVITY, STRENGTH 
OF MATERIALS, WATER WHEELS, HYDRAULICS, HYDROSTATICS, PNEU- 
MATICS, STATICS, DYNAMICS, GUNNERY, HEAT, WINDING EN- 
GINES, TONNAGE, SHOT, SHELLS, &C. 

STEAM AND THE STE AM-ENGINE j 

COMBUSTION, WATER, CABLES AND ANCHORS, FUEL, AIR, GUNS, &C., &C, 

TABLES OF THE WEIGHTS OF METALS, PIPES, &C. 

MISCELLANEOUS NOTES, AND EXERCISES, 

&c., &:c. 

/ 

BY CHARLES H. HASWELL. 

CHIEF ENGINEER U. S. NAVY. 



All examination of facts is the foundation of science. 



NEW-YORK: 

HARPER 6c BROTHERS, 82 CLIFF- STREET. 



1844. 



]m-^ f^' 



Kntfired, accordii^ig to Act of Con-zrps^, in the year 1844, by • 

IIaki'kr & Bkothkrs, 

In thp Clerk's Office of the Southern District of New-York. 



ERRATA. 
Page 140, 27th line, for " common," read cannon. 



" 143, 12th line, for *y 10x64.33," read v/lOxbISC 
" 142, 16th and 17th lines, omit *'and that product, attain, by 
the velocity in feet per second." "^ 

Page 142, 20th line, omit *' x5=:7500." 
" 158, 15th line, for " 594,000," read 59,400. 
" 179, 26th line, for " 5}" read 4,V. \ 

" 218, last line, after "horses' power," read, witU plain cyli»^ 
drical boilers. I 

Page 226, 17th line, for *' Picton," read Pictou. 
*' 240, 19th line, for " brushes," read bushes. i 

" 248, in cohimn 2d, insert i opposite to '* 0.211." 
•* 251, 3d line from bottom, for '* 176.7149," read 1767 145 1^ 
and in bottom line, for "460.6957," read 460.6947. j 

Page 260, last line, for " 92000," read 9200. 






^y^2Z~ 4^::^^^^;^. 



TO 



CAPTAIN ROBERT F.STOCKTON, 

U. S. NAVY, 

AS A TRIBUTE TO THE LIBERALITY AND ENTERPRISE HE HAS 

EVINCED IN HIS PATRONAGE OF MECHANICAL SCIENCE, 

THIS EDITION IS, WITH PERxMISSION, 

BY 

THE AUTHOR. 

Washington^ Aug. 1, 1843. 



PREFACE. 



The following work is submitted to the Engineers and 
Mechanics of the United States by one of their number, 
who trusts that it will be found a convenient summary for 
reference to Tables, Results, and Rules connected with 
the discharge of their various duties. 

The Tables have been selected from the latest and best 
publications, and information has been sought from various 
sources, to render it useful to the Operative Engineer, Me- 
jchanic, or Student. 

The want of a work of this description in this country 
has long been felt, and this is peculiarly fitted to supply 
that want, in consequence of the adaptation of its rules to 
the metals, woods, and manufactures of the United States. 

Having for many years experienced inconvenience for 
the want of a compilation of tables and rules by a Prac- 
tical Mechanic, together with the total absence of units for 
the weights and strengths of American materials, I was in- 
duced to attempt the labour of a compilation and the neces- 
sary experiments to furnish this work. 

The proportions of the parts of the steam-engine and boilers 
will be found to differ in most instances materially from the 
English authorities ; but as they are based upon the actual 
results of the most successful experience, I do not hesitate 
to put them forth, being well assured that an adherence to 
them will ensure both success and satisfaction. 

A2 



Vi PREFACE. 

The sources of information from which I have principally- 
compiled are Adcock, Grier, Gregory, the Library of Use- 
ful Knowledge, and the Ordnance Manual ; and to the la- 
bours of the authors of these valuable works I freely ac- 
knowledge my indebtedness. 

In my own efforts, I have been materially assisted by the 
officers of the West Point Foundry Association, who liber- 
ally furnished me wdth the means of making such experi- 
ments as were considered necessary, and to the Engineer 
of that establishment, Mr. B. H. Bartol, I am indebted for 
much valuable information and assistance. 

To the Young Engineer I would say, cultivate a knowl- 
edge of physical laws, without which, eminence in his pro- 
fession can never be securely attained ; and if this volume 
should assist him in the attainment of so desirable a result, 
the object of the author will be fully accomplished. 



We have seen a proof copy of Has well's Engi- 
neers' AND Mechanics' Pocket Book, and approve of 
its design and the subjects treated of: a work of this 
description has long been wanted, and we confidently 
express a conviction of its usefulness and appHcation, 
which in extent exceeds that of any work of its class 
with which we are acquainted. 



GOUVERNEUR KeMBLE, 

William Kemble, 
Robe'rt p. Parrott, 
B. H. Bartol, 
James P. Allaire, 

B. R. M'Ilvaine, 
Horatio Allen, 

C. E. Detmold, 

Charles W. Hackley, 

Hogg & Delamater, 
Stillman & Co., 

T. F. Secor & Co., 

Brown & Bell, 
Smith & Dimon, 

Merrick & Towne, 



I We5^ Point Foundry 
I Association, N. Y. 

I Allaire Works, N. Y, 

> Civil Engineers. 

{ Professor of Mathematics, 
\ Columbia College, N. Y. 

Phxnix Foundry, N. Y. 

Novelty Works, N. Y. 

S Steam- Engine Manufac- 
turers, N. Y. 

\ Shipbuilders, N. Y. 

SSouthwark Foundry f 
Philadelphia. 



CONTENTS. 



Notation 11 

Explanations OF Characters. • . 12 

U. S. WEIGHTS AND MEASURES. 

Measures of Length 13 

Measures of Surface 14 

Measures of Capacity 14 

Measures of Solidity 15 

Measures of Weight 15 

Miscellaneous 16 

Measures of Value 16 

Mint Value of Foreign Coins 16 

FOREIGN WEIGHTS AND MEASURES. 

Measures of Length 17 

Measures of Surface 18 

Measures of Capacity 18 

Measures of Solidity 19 

Measures of Weight 20 

Scripture and Ancient Meas- 
ures 21 

Table for finding the Distances of 

ODjects at Sea 21 

Reduction of Longitude 22 

Vulgar Fractions 23 

Decimal Fractions 25 

Diwdecimals 30 

Rule of Three 31 

Compound Proportion 32 

Involution 32 

Evolution 32 

Arithmetical Progression 34 

Geometrical Progression 35 

Permutation 36 

Combination 36 

Position 36 

Fellowship 37 

Alligation 38 

Compound Interest 38 

Discount 39 

Equation of Payments 39 

Annuities • • • • 40 

Perpetuities 41 

Chronological Problems 41 

To find the Moon's Age 42 

Table of Epacts, Dominical Letters, 

&c 42 

Promiscuous Questions 43 

GEOMETRY. 

Definitions 46 



CONIC SECTIONS. p^gg 

Construction of Figures 48 

Definitions 54 

To construct a Parabola 54 

To construct a Hyperbola 54 

Ellipse 55 

Parabolas 56 

Hyperbolas 57 

MENSURATION OF SURFACES. 

Triangles, Trapeziums, and Trape- 
zoids 59 

Regular Polygons 60 

Regular Bodies, Irregular Figures, 

and Circles 61 

Arcs of a Circle 62 

Sectors, Segments, and Zones 63 

Ungulas and Ellipses 64 

Parabolas and Hyperbolas 65 

Cylindrical Rings and Cycloids 66 

Cylinders, Cones, Pyramids, Spheres, 

and Circular Spindles 67 

By Mathematical Formulce. 
Lines of Circle, Ellipse, and Para- 
bola 67 

Areas of Quadrilaterals, Circle, Cyl- 
inder, Spherical Zone or Segment, 
Circular Spindle, Spherical Tri- 
angle, or any Surface of Revolu- 
tion 68 

Capillary Tube 69 

Useful Factors 69 

Examples in Mensuration 70 

Areas of the Segments of a 

Circle 72 

Lengths of Circular Arcs 75 

To find the Length of an Elliptic 

Curve 77 

Lengths op Semi-elliptic Arcs . . 78 

Areas of THE Zones OF A Circle. 80 

MENSURATION OF SOLIDS. 

Of Cubes and Parallelopipedons .... 81 

" Regular Bodies 81 

" Cylinders, Prisms, and Ungulas 81 

" Cones and Pyramids 82 

*' Wedges and Prismoids 83 

" Spheres 83 

" Spheroids 84 

" Circular Spindles 85 

" Elliptic Spindles 85 

" Parabolic Conoids and Spindles . 86 



CONTENTS. 



IX 



Page 
Of Hyperholoids and Hyperbolic Co- 
noids 86 

" Cylindrical Rings 87 

By Mathematical FormulcB 87 

Cask Gauging- 88 

Examples in Mensuration 89 

Areas OF Circles 91 

Circumferences OF Circles 95 

Squares, Cubes, AND Roots 99 

To find the Square of a J^umber 

above 1000 116 

To find the Cube or Square Root of 
a higher JsTumber than is contain- 
ed in the Table 118 

To find the Cube of a JSTumber above 

1000 118 

To find the Sixth Root of a JVumber 118 
To find the Cube or Square Root of 
a JSTumber consisting of Integers 

and Decimals 119 

Sides of Equal Squares 120 

Plane Trigonometry 123 

Oblique-angled Triangles 124 

Natural Sines, Cosines, and Tan- 
gents 125 

Sines and Secants 127 

MECHANICAL POWERS. 

Lever 128 

Wheel and Axle 130 

Inclined Plane 131 

Wedge 133 

Screw 133 

Pulley 135 

CENTRES OF GRAVITY. 

Surfaces 137 

Solids 138 

Gravitation 139 

Promiscuous Examples 142 

Gravities of Bodies . . .' 143 

Specific Gravities 143 

Proof of Spirituous Liquors 144 

Table of Specific Gravities 145 

STRENGTH OF MATERIALS. 

■Cohesion 148 

Transverse Strength 149 

Deflexion 156 

Journals of Shafts 158 

Gudgeons and Shafts 160 

Teeth of Wheels 161 

Velocity of Wheels 162 

Strength of Wheels 163 

General Explanations concerning 
Wheels 164 

Horse Power 165 

ANIMAL STRENGTH. 

Men 165 

Horses 166 



HYDROSTATICS. 

Pagft 

Of Pressure 167 

Construction of Banks 168 

Flood Gates 168 

Pipes 168 

Hydrostatic Press 169 

HYDRAULICS AND HYDRODYNAMICS. 

Of Sluices 170 

Of Vertical Jipertures or Slits 170 

Of Streams and Jets 171 

Velocity of Streams 171 

To find the Velocity of Water run- 

7iing through Pipes 1 72 

Waves 172 

Table showing the Head necessary 

to overcome the Friction of Water 

in Horizontal Pipes 173 

General Rules 174 

Table of the Rise of Watar in Rivers 175 

Water Wheels 176 

To find the Power of a Stream 176 

Barker's Mill 178 

To find the Centre of Gyration of a 

Water Wheel 179 

JVotes 179 

PNEUMATICS. 

Weighty Elasticity, and Rarity of 
Air 180 

Measurement of Heights by means 

of the Barometer 181 

Velocity and Force of Wind 181 

STATICS. 
Pressure of Earth against Walls . . 182 

Dynamics 183 

Pendulums 185 

Centre of Gyration 186 

Centres of Percussion and Os- 
cillation 188 

Central Forces 190 

Fly Wheels 192 

Governors 192 

Gunnery 193 

Friction 194 

HEAT. 

Communication of Caloric 196 

Radiation of Caloric 197 

Specific Caloric 197 

Evaporation 200 

Congelation and Liquefaction 200 

Distillation 200 

Miscellaneous 201 

Melting Points of Alloys 202 

Gunpowder 203 

Dimensions of Powder Barrels .... 203 

Light 204 

Tonnage 204 



CONTENTS. 



Page 

Piling op Balls and Shells 206 

Weights and Dimensions of Balls 

and Shells 207 

Winding Engines 208 

Fraudulent Balances 209 

Measuring of Timber 210 

Steam 211 

Steam acting- Expansively 214 

STEAM-ENGINE. 

Condensing' Engines 216 

Boilfirs 217 

JVon- condensing Engines 218 

Boilers 218 

General Rules 219 

Saturation in Marine Boilers 219 

Sjnoke Pipes, or Chimneys 220 

Belts 220 

Power of Engines 221 

To find the Volume the Steam, of a 

Cubic Foot of Water occupies 222 

To find the Power of an Engine ne- 
cessary to raise Water 222 

To find the Velocity necessary to 
Discharge a Given Quantity of 
Water 223 

COMBUSTION. 

Fuel 224 

.Analysis of Fuels 225 

.Anthracite 226 

Charcoal 226 

Coke 226 

Water 227 

Motion of Bodies in Fluids 229 

Air 232 

Dimensions and Weight of Guns, 

Shot, and Shells, U. S. Army. 233 
Weia-ht and Dimensions of Leaden 

Bills 234 

Expansion of Shot 234 

Weight and Dimensions of Shot 234 

Dimensions and Weight of Guns, 
Shot, and Shells, U. S. Navy. . 236 

Penetration of Shot and Shells 237 

Penetration of Shells 238 

MISCELLANEOUS. 

Recapitulation of Weights of Vari- 
ous Substances 239 

Weights of a Cubic Foot of Various 
Substances 239 

Slating 239 

Capacity of Cisterns 240 



Page 

Compositions 240 

Sizes ofJ^uts 240 

Screws 241 

Strength of Copper 241 

Digging 241 

Coal Gas 241 

Mcohol 242 

Composition Sheathing J^ails 242 

Cement 242 

Brown Mortar 242 

Bricks, Laths, &c. 243 

Hai/ 243 

Hills in an ^cre of Ground 243 

Displacement of Vessels 243 

Weight of Lead Pipe 244 

TiJ...-- 244 

Relative Prices of American 

Wrought Iron 345 

Power required to Punch Iron and 

Copperplates 245 

IRON. 

Weight of Square Rolled- Iron 246 

Weight of Round Rolled Iron 247 

Weight of Flat Rolled Iron 248 

Weight of a Square Foot of Cast 
and Wrought Iron, Coppery and 

Lead....:. 2,51 

Cast Iron 251 

Weight of Cast Iron Pipes 252 

Weight of a Square Foot 254 

Weight of Cast Iron and Lead Balls 255 
Weight of Copper Rods or Bolts • . • • 256 
Weight of Riveted Copper Pipes .... 256 

Copper 257 

Lead 257 

Brass 257 

Cables and Anchors 258 

Cables 259 ' 

Tables of Hemp Cables and Ropes . . 259 ' 
To ascertain the Strength of Cables. 259 
To ascertain the Weight of Manilla 

Ropes and Hawsers • • 259 

To ascertain the Strength of Ropes. 260 
To ascertain the Weight of Cable-laid 

Ropes 260 

To ascertain the Weight of Tarred 

Ropes and Cables 260 

Blowing Engines 261: 

Dimensions of a Furnace, Engine, 
&c 2611 

MISCELLANEOUS NOTES. 

On Materials 262 

Solders, Cements, and Paints 263 

Paints, Lackers, and Sta,ining 26'* 



NOTATION. 



1 = 

2 = 

3 = 

4 = 

5 = 

1- 
S-. 
9 = 

10: 
20: 
30: 
40: 
50: 
60: 
70: 
80: 

90 
100 



1 1. 

rll. 
= 111. 
= IV. 

zV. 
= VI. 
= VIL 
= VIII. 

zIX. 

= X. 
= XX. 
= XXX. 

= XL. 
= L. 
= LX. 
= LXX. 

=:LXXX. 

= xc. 



500 = D, or 10. 



1,000 = 



M, or CIO. 
MM. 



i As often as a character is repeated, 
\ so many times is its value repeated. 

( A less character before a greater 
} diminishes its value, as IV = I from 
( V, or 1 subtracted from 5 = 4. 

c A less character after a greater in- 
\ creases its value, as XI = X+I, or 

ho+i = ii. 



c For every annexed, this be- 
\ comes 10 times as many. 
I For every C and 0, placed one at 
\ each end, it becomes 10 times as 
( many. 



2,000 __ 
5,000=:V, or 100. 

6,000 = VI. 

10,000 =% or CCIOO. 

50,000 ^T, or 1000. 

60,000 :==LX. 

100,000 = 0, or CCCIOOO. 
1,000,000 =M>r CCCCIOOOO. 
2,000,000 = MM. 

Examples.— 1840, MDCCCXL. 
18560, XVIIIDLX. 



A bar, thus — , over any 
> number, increases it 1000 
times. 



12 



EXPLANATION OF CHARACTERS. 



EXPLANATIONS OF THE CHARACTERS USED IN THE FOLLOWING 
TABLES AND CALCULATIONS. 

= Equal to, as, 12 inches = 1 foot, or 8x8 = 16x4. 

+ Plus, or more, signifies addition ; as, 4+6+5 = 15. 

— Minus, or less, signifies subtraction ; as, 15—5 — 10. 

X Multiplied hy, or into, signifies multiplication ; as, 8x9 r=: 72 

~ Divided hy, signifies division ; as, 72-:-9 = 8. 

•'• so is I Proportion ; as, 2 : 4 : : 8 : 16 ; that is, as 2 is to 4 so is 8 
:'-to S ^o\Q. 
V Prefixed to any number signifies that the square root of that 

number is required ; as, ^16=:4; that is, 4x4=r 16. 
^ Signifies that the cube root of that number is required • as 
^64 1=4; that is, 4X4X4 = 64. 

2 added to a number signifies that that number is to be squared • 

thus, 42 means that 4 is to be multiplied by 4. 

3 added to a number signifies that that number is to be cuhed • 

thus, 43 IS = 4 X 4 X 4 == 64. The power or number of times 
a number is to be multiplied by itself is shown by the num- 
ber added ; as, 2 3 4 5^ &c. 

~~ The har si gnifies that the numbers are to be taken together ; 

as, 8— 2+6 r= 12, or 3x5+3 = 24. 
. Decimal point, signifies when prefixed to a number, that that 
nunaber has a unit (1) for its denominator ; as, .1 is-i, .155 
. ^^ ToVo ' ^^• 

CO Signifies difference, and is placed between two quantities when 
it is not evident which of them is the greater. 

° Degrees, ' minutes, " seconds, '" thirds. 

< Signifies angle. 

^~^ ^~-^' ^— -J <^c., denote inverse powers of a, and are equal 

a^ a^ a^ 
7 Is put between two quantities to express that the former is 

greater than the latter ; as, alb, reads a greater than b. 
L Signifies the reverse ; as, a Z, b, reads a less than b. 
( ) Parentheses are used to show that all the figures within them 

are to be operated upon as if they were only one : thus 

(3+2)x5 = 25. 
p is used to express the ratio of the circumference of a circle to 

its diameter = 3.1415926, &c. 
A A' a:' K!" signifies A, A prime, A second, A third, &c. 
dXd, a.d, or ad, signifies that a is to be multiplied by d. 

To ascertain the value of a decimal of a unit, see Reduction of 
Decimals, page 28. 

Note. The degrees of temperature used in this work are those of 
Fahrenheit. 



WEIGHTS AND MEASURES. 13 



UNITED STATES' WEIGHTS AND MEASURES. 



12 inches 


— 1 foot. 


3 feet 


= 1 yard. 


5i yards 


= 1 rod. 


40 rods 


= 1 furlong 



8 furlongs = 1 mile. 



Measures of Length. 



36= 3. 

198= 16^= 5i 
7920= 660 = 220 = 40. 
63360 = 5280 =1760 =320 = 8. 
The inch is sometimes divided into 3 larley corns, or 12 lines. 
A hair's breadth is the 48th of an inch. 

Gunter^s Chain. 
7.92 inches =: 1 link. 
100 links = 4 rods, or 22 yards. 

Ropes and Cables. 
6 feet = 1 fathom. 
120 fathoms = 1 cable's length. 

Geographical and Nautical Measure. 
1 degree of a great circle of the earth =: 69.77 Statute miles. 
1 mile = 2046.5 yards. 

Log Lines. 

1 knot = 51.1625 feet, or 51 feet lj+ inches. 

1 fathom = 5.11625 feet, or 5 feet 1^+ inches. 

Estimating a mile at 61391 feet, and using a 30" glass. If a 28" 

glass is used, and eight divisions, then 

1 knot = 47 feet 9 + inches. 
1 fathom =z 5 feet llf inches. 
The line should be about 150 fathoms long, having 10 fathoms 
between the chip and first knot for stray line. 

'Note.— Bowditch gives Q\'20 feet in a sea miUy whichy if taken as the lengthy will 
make the divisions 51 feet and 5 I-IO feet. 

Cloth. 

1 nail = 2i inches = ^^th of a yard. 

1 quarter = 4 nails. 

5 quarters = 1 ell English 

Pendulums. 

6 points =: 1 line. 

12 lines =: 1 inch. 

Shoemakers\ 

No. 1 is 4i inches in length, and erery succeeding number is I of 
an inch. 

^ There are 28 divisions, in tw^o series of numbers, viz., from 1 to 
"13, and 1 to 15. 

B 



14 



WEIGHTS AND MEASURES. 



Circles, 
60 seconds = 1 minute. 
60 minutes = I degree. 
360 degrees = 1 circle. 
1 day is ... . 
1 minute is . 



3600 
1296000 : 



: 21600. 



.002739 of a year. 
.000694 of a day. 



Miscellaneous, 



1 palm = 3 inches. 
1 hand = 4 inches. 



1 span : 
1 metre : 



: 9 inches. 
: 3.28174 feet. 



The standard of measure is a brass rod, which, at the tempera- 
ture of 32° Fahrenheit, is the standard yard. 

The standard yard of the State of New-York bears, to a pendulum 
vibrating seconds in vacuo, at Columbia College, the relation of 
1.000000 to 1.086141 at a temperature of 32° Fahrenheit. 

1 yard is 000568 of a mile. 

1 inch is 0000158 of a mile. 



Measures of Surface, 

144 square inches = 1 square foot. I 
9 square feet =z \ square yard. | 



Inches. 

1296 



Rods. 



Land. 
30J square yards r=: 1 square rod. 
40 square rods = 1 square rood, 
4 square roods ) _ ^ ^^^^ 
10 square chains 5 
640 acres = 1 square mile, 

E.— 208.710321 /ee^, 69.5701 yards, or 220 by 198 /eei square = 1 acre. 

Paper, 
24 sheets = 1 quire. I sheets. 

20 quires = 1 ream. | 480. 



Roods. 



Yards. 

1210. 

4840 = 160. 
3097600 = 102400 = 2560. 



Note.— 2 



Cap 

Demy . 

Medium 

Royal . 

Super-royal 

Imperial 

Elephant 



13 X16 inches 

19ixi5i- " 

22 X18 " 

24 X19 " 

27 X19 " 

29 X21i " 

27ix22i *' 



Draiving Paper. 



Columbier . 33} X 23 inches. 

Atlas . . 33 X 26 

Theorem . 34 x 28 

Doub. Elephant, 40 x 26 

Antiquarian . 52 x 31 

Emperor . 40 x 60 

Uncle Sam . 48 x 120 



Measures of Capacity, 

Liquid. 
4 gills = 1 pint. 
2 pints =z 1 quart. 
4 quarts = 1 gallon. 
The standard gallon measures 231 cubic inches, and contains 



Pints. 



Gills. 
8. 

32 = 8 



WEIGHTS AND MEASURES. 



15 



8 3388822 avoirdupois pounds, or 58372.1754 troy grains of distilled 
water at 39° 83 Fahrenheit ; the barometer at 30 inches. 

The gallon of the State of New-York contains 221.184 cubic inch- 
es, or 8 pounds of pure water at its maximum density. 



The Imperial gallon (British) contains 277.274 cubic inches. 



Pints. Quarts. Gallons. 

8. 
16 = 8. 
64 — 32 = 8. 
which contains 2150.42 



Dry. 
2 pints = 1 quart. 
4 quarts = 1 gallon. 
2 gallons = 1 peck. 
4 pecks = 1 bushel. 
The standard bushel is the Wincheste\ 
cubic inches, or 77.627413 lbs. avoirdupois of distilled water at its 
maximum density. 

Its dimensions are 18i inches diameter inside, 19^ inches out- 
side, and 8 inches deep ; and when heaped, the cone must not be 
less than 6 inches high. 

The bushel of the State of New- York contains 80 lbs. of pure water 
at its maximum density, or 2211.84 cubic inches. 



Measures of Solidity. 



1728 cubic inches r=z l foot. 
27 cubic feet =. 1 yard. 



Inches. 

46656. 



Miscellaneous, 
1 chaldron = 36 bushels, or . 57.25 cubic feet. 

Dry gallon of New- York . . 276.48 cubic inches. 
1 perch of stone . . . . 24.75 cubic feet. 

Measures of Weight 

Avoirdupois. 
16 drachms = 1 ounce. 
16 ounces •=: 1 pound. 
112 pounds r= 1 cwt. 
20 cwt. = 1 ton. 

1 lb. = 14 oz. 11 dwt. 16 gr. troy. 
The standard avoirdupois pound is the weight of 21^7015 cubic 
inches of distilled water weighed in air, at the tempdfature of the 
maximum density (3^^.83), the barometer being at 30 inches. 

Troy. 



Ounces. 



Pounds. 



Drachms. 

256. 
28672 = 1792. 
573440 = 35840 = 2240. 



24 grains = 1 dwt 
20 dwt. = 1 ounce. 
12 ounces = 1 pound. 

^ Apothecaries. 

20 grains ' = 1 scruple. 

3 scruples = 1 drachm. 

8 drachms = 1 ounce. 
12 ounces = 1 p?^nd. 



Grains. 

480. 
5760 



:240. 



Grains. Scruples. Drachms. 

60. 
480 = 24. 
5760 — 288 = 96. 



16 



WEIGHTS AND MEASURES. 

Diamond. 
16 parts — 1 grain = 0.8 troy grains. 
4 grains = 1 carat = 3.2 " 



7000 troy grains = 1 lb. avoirdupois. 

175 troy pounds = 144 lbs. " 

175 troy ounces = 192 oz. " 

437i troy grains = 1 oz. " 

1 troy pound = .8228+ lb. " 

Miscellaneous, 

1 cubic foot of anthracite coal from 50 to 55 lbs. 
1 cubic foot of bituminous coal from 42 to 55 lbs. 
1 cubic foot Cumberland coal = 53 lbs. 
1 cubic foot charcoal . — 18.5 '' (hard wood). 
1 cubic foot charcoal . = 18. " (pine wood). 
1 bushel bituminous coal — 80 " 
1 stone . . . . = 14 " 
Coals are usually purchased at the conventional rate of 28 bush- 
els (5 pecks) to a ton. 

Measures of Value. 

1 eagle = 258 troy grains. 
1 dollar = 412.5 " 
1 cent — 168 " 
The standard of gold and silver is 900 parts of pure metal, and 100 
of alloy, in 1000 parts of coin. 

Relative Mint Value of Foreign Gold Coins, 



By Laiv of Congress^ August, 1834. 



Brazil. 


1 Johannes 




1 Dobraon . 




1 Dobra 




1 Moidore . 




'k Crusado . 


England. 


^Guinea . 
^Sovereign 




France. 


1 Double Louis (1786) 




1 Double Napoleon . 


Colombia. 


1 Doubloon 


Mexico. 


1 Doubloon 


Portugal. 


1 Dobraon . 




1 Dobra 




1 Johannes 




1 Moidore . 




1 Milrea . 


Spain. 


1 Doubloon (1772) 




1 Doubloon (1801) 




1 Pistole . 



Weight. 
Dwt. Gr. 
18 

34 12 
18 06 

6 22 
16i 

5 9i 

5 3^ 
10 11 

8 7 
17 8^ 

17 8i 
34 12 

18 6 
18 

6 22 
19^ 

17 8i 
17 9 
4 3i 



Value. 

$17,068 

32.714 

17.305 

6.560 

.638 

5.116 

4.875 

9.694 

7.713 

15.538 

15.538 

32.714 

17.305 

17.068 

6.560 

.780 

16.030 

* 15.538 

3.883 



23.2 grains of pure gold = $1.00. 
United States Eagle preceding 1834,' $10,668. 



FOREIGN WEIGHTS AND MEASURES. 



17 



Mint Value of Foreign Coins. 



England. 
France. 



Austria. 



Prussia. 
Russia. 



Sweden. 



1 Shilling . 

5 Francs 

1 Sous .... 

1 Crown, or rix dollar . 

1 Ducat 

1 Ducat 

1 Ducat = 10 roubles . 

1 Rouble 

1 Ducat 

1 Rix dollar . 



$0,244 
0.935 
0.0093 
0.97 

2.22 

2.202 

7.724 

0.743 

2.19 

1.08 



The relative value of gold and silver is as 1 to 15 Ji. 



r.A 



y/ Measures, of Length. ^ 

Yardis^the length of a pendulum vibrating seconds in vacuo in Lon 
donfaf the level of the sea ; measured on a brass rod, at the tem 
perature of 62° Fahrenheit, =39.1393 Imperial inches. 



# 



French. Old System.— 1 Line =: 
1 
1 
1 
1 
1 
1 
JVew System. — 1 
1 
1 
1 
1 
1 



Austrian 
Prussian 
Swedish 
Spanish 



12 points . 
Inch = 12 lines . 
Foot = 12 inches . 
Toise = 6 feet . 
League =: 2280 toises . 
League = 2000 toises . 
Fathom =: 5 feet. 
Millimetre 
Centimetre 
Decimetre . 
Metre .... 
Decametre . 
Hecatometre 
Foot .... 
Foot . . . 
Foot .... 
Foot .... 
League (common) 



0.08884 U. S. inches. 

= 1.06604 

= 12.7925 

= 76.7550 " 

(common), 
(post). 

= .03938 U. S. inches. 

= .39380 

= 3.93809 

= 39.38091 

= 393.80917 

= 3938.09171 " 

= 12.448 " 

= 12.361 

= 11.690 

= 11.034 

= 3.448 U. S. miles. 



Table showing the relative length of Foreign Measures compared 
with British. 



Plcuies. 




Measures. 


Inches. 


Places. 




. 


Measures. 


Inches. 


Amsterdam . 


Foot 


11.14 


Malta . . . 


Foot 


11.17 


Antwerp . . 


" 


11.24 


Moscow 






" 


13.17 


Bavaria 




" 


11.42 


Naples . 






Palmo 


10.38 


Berlin . 




u 


12.19 


Prussia 






Foot 


12.35 


Bremen . 




" 


11.38 


Persia . 






Arish 


38.27 


Brussels 




u 


11.45 


Rhineland 






Foot . 


12.35 


China . 




" Mathematic. 
" Builder's 
" Tradesman's 


13.12 
12.71 
13.32 


Riga . . 
Rome . 
Russia . 








10.79 
11.60 
13.75 


" 




" Surveyor's 


12.58 


Sardinia 






Palmo 


9.78 


Copenhage 


a . 


" 


12.35 


Sicily . 








9.53 


Dresden 




' " 


11.14 


Spaiij . 






Foot 


11.12 


England 




" 


12.00 


" 






Toesas 


66.72 


Florence 




Braccio 


21.60 


" 






Palmo 


8.34 


France . 




Pi^d de Roi 
Metre 


12.79 
39.381 


Strasburgh 
Sweden 






Foot 


11.39 
11.69 


Geneva . 




Foot 


19.20 


Turin . 






«' 


12.72 


Genoa . 




Palmo 


9.72 


Venice . 






(( 


13.40 


Hamburgh 




Foot 


11.29 


Vienna . 






u 


12.45 


Hanover 




(( 


11.45 


Zurich . 






it 


11.81 


Leipsic . 




" 


11.11 


Utrecht 






" 


10.74 


Lisbon . 




" 


12.96 


Warsaw 






*' 


14.03 


" , 




Palmo 


8.64 









18 



FOREIGN WEIGHTS AND MEASURES. 



Table showing the relative length of Foreign Road Measures com- 
pared with British. 



Places. 




Measures. 


Yards. 


Placet. 


Measures. 


Tarda. 


Arabia . 


Mile 


2148 


Hungary . . 


Mile 


9113 


Bohemia 




" 


10137 


Ireland . . . 


" 


3038 


China . 




Li 


629 


Netherlands . 


" 


1093 


Denmark 




Mile 


8244 


Persia . . . 


Parasang 


6086 


England . 




" Statute 


1760 


Poland . . . 


Mile, long 


8101 




" GeogTaph. 


2025 


Portugal . . 


League 


6760 


Flanders 




" 


6869 


Prussia . . . 


Mile 


8468 


France . 




League, marine 


6075 


Rome . . . 


" 


2025 






" common 


4861 


Russia . . . 


Verst 


1167 


ii 




" post 


4264 


Scotland . . 


Mile 


1984 


.Germany 




Mile, long 


10126 


Spain . . . 


League, com. 


7416 

11700 

9153 


Hamj3,ttrgh 
Hanover 




a * 


8244 
1T559 


Sweden . . 
•Switzerland . 


Mile 


Holland . 




" 


6395 


Turkey . . . 


Berri 


1826 



Measures of Surface, 



French. Old System.— 1 Square Inch 

1 Arpent (Paris) . 
1 Arpent (woodland) 
JSTew System. — 1 Jire . 
1 Decare 
1 Hecatare . 
1 Square Metre . 

1 Are . 



= 1.1364 U. S. inches. 
= 900 square toises. 
= 100 square royal perches. 
= 100 square metres. 
= 10 ares. 
= 100 ares. 

= 1550.85 square inches, 
or 10.7698 square feet. 
= 1076.98 



Table showing the relation of Foreigv. 3feasures of Surface compa- 
red with British. 



Amsterdam 
Berlin. 

Canary Isles 
England . 
Geneva . 
Hamburgh 
Hanover 
Ireland . 
Naples . 



Sq. yards. 




Acre 
Moggia 



Places. 


Measures. 


Sq. yards. 


Portugal . . 


Geira 


6970 


Prussia . . . 


Morgen 


3053 


Rome .... 


Pezza 


3158 


Russia . . . 


Dessetina 


13066.6 


Scotland . . 


Acre 


6150 


Spain . . . 


Fanegada 


5500 


Sweden . . 


Tunneland 


5900 


Switzerland . 


Faux 


7855 


Vienna . . . 


Joch 


6889 


Zurich . . . 


Common acre 


3875.6 



Measures of Capacity. 



British. The Imperial gallon measures 277.274 cubic inches, containing 10 lbs. 
avoirdupois of distilled water, weighed in air, at the temperature ot 
62^, the barometer at 30 inches. 
For Grain. 8 bushels = 1 quarter. 

1 quarter = 10.2694 cubic feet. 
CkfaL or heaped measure. 3 bushels = 1 sack. 

12 sacks = 1 chaldron. 
Imperial bushel = 2218.192 cubic inches. 

Heaped bushel, 19^ ins. diam., cone 6 ms. high = 2815.4872 cub. ins. 
1 chaldron = 58.658 cubic feet, and weighs 3136 pounds. 
1 chaldron (Newcastle) = 5936 pounds. tt c , u ;«« 

French. J^ew System.-l Litre = 1 cub. decimetre, or 61.074 U. S. cub- ms. 

Old System. - 1 Boisseau = 13 litres = 793.964 cub. ms., or 3.43 galls. 
1 Pinte = 0.931 litres, or 56.817 cubic inches. 
Spanish. 1 Wine Arroba = 4.2455 gallons. 

1 Fanega (common measure) = 1.593 bushels. 



FOREIGN WEIGHTS AND MEASURES. 



19 



Table showing the relative Capacity of Foreign Liquid Measures 
compared with British. 



Places. 


Measnres. 


Cub. Inch. 


Places. 




Measures. 


Cub.Inch. 


Amsterdam 


Anker 


2331 


Naples . . . 


Wine Barille 


2544 


" 


. . Stoop 


146 


" 






Oil Stajo 


1133 


Antwerp 


. . " 


194 


Oporto 






Almude 


1555 


Bordeaux 


. . Barrique 


14033 


Rome 






Wine Barille 


2560 


Bremen . 


. . Stubgens 


194.5 


" 






Oil 


2240 


Canaries 


. . Arrobas 


949 


u 






Boccali 


80 


Constantino 


pie Almud 


319 


Russia 






Weddras 


752 


Copenhagei 


I . Anker 


2355 








Kunkas 


94 


Florence 


. Oil Barille 


1946 


Scotland 






Pint 


103.5 




. Wine " 


2427 


Sicily 






Oil Caffiri 


662 


France . 


. Litre 


61.07 


Spain 






Azumbres 


22.5 


Geneva . 


. Setier 


2760 








Quartillos 


30.5 


Genoa . 


. Wine Barille 


4530 


Sweden 






Eimer 


4794 


" 


. Pinte 


90.5 


Trieste 






Orne 


4007 


Hamburgh 


. Stubgen 


221 


Tripoli 






Mattari 


1376 


Hanover 


" 


231 


Tunis 






Oil " 


1157 


Hungary 


Ehner 


4474 


Venice 






Secchio 


628 


Leghorn . 


. Oil Barille 


1942 


Vienna 






Eimer 


3452 


Lisbon . 


Almude 


1040 


'* 






Maas 


86.33 


Malta . . 


. Caffiri 


1270 









Table showing the relative Capacity of Foreign Dry Measures com- 
pared with British. 



Places. 


Measures. 


Cub.Inch. 


Places. 




Measures. 


Cub.Inch. 


Alexandria 


Rebele 


9587 


Malta . . . 


Salme 


16930 


" . . 


Kislos 


10418 


Marseilles 




Charge 


9411 


Algiers . . . 


Tarrie 


1219 


Milan . 




Moggi 


8444 


Amsterdam . 


Mudde 


6596 


Naples . 




Tomoli 


3122 


" . . 


Sack 


4947 


Oporto . 




Alquiere 


1051 


Antwerp . . 


Viertel 


4705 


Persia . 




Artaba 


4013 


Azores . . . 


Alquiere 


731 


Poland . 




Zorzec 


3120 


Berlin . . . 


Scheffel 


3180 


Riga . . 




Loop 


3978 


Bremen . . . 


" 


4339 


Rome . 




Rubbio 


16904 


Candia . . . 


Charge 


9288 


" 




auarti 


4226 


Constantinople 


Kislos 


2023 


Rotterdam 




Sach 


6361 


Copenhagen . 


Toende 


8489 


Russia . 




Chetwert 


12448 


Corsica . . . 


Stajo 


6014 


Sardinia 




Starelli 


2988 


Florence . . 


Stari 


1449 


Scotland 




Firlot 


2197 


Geneva . . . 


Coupes 


4739 


Sicily . 




Salme gros 


21014 


Genoa . . . 


Mina 


7382 


" 




" generale 


16886 


Greece . . . 


Medimni 


2390 


Smyrna. 




Kislos 


2141 


Hamburgh . . 


Scheffel 


6426 


Spain . 




Catrize 


41269 


Hanover . . 


Malter 


6868 


Sweden . 




Tunnar 


8940 


Leghorn . . . 


Stajo 


1501 


Trieste . 




Stari 


4521 


" ... 


Sacco 


4503 


Tripoli . 




Caffiri 


19780 


Lisbon . . . 


Alquiere 


817 


Tunis . 




u 


21855 


" ... 


Fanega 


3268 


Venice . 




Stajo 


4945 


Madeira . . . 


Alquiere 


684 


Vienna . 




Metzen 


3753 


Malaga . . . 


Fanega 


3783 









Measures of Solidity. 

French. 1 Cubic Foot = 2093.470 U. S. inches. 

Decistre = 3.5375 cubic feet. 

Steve (a cubic metre) . . . . = 35.375 " 

Decastere = 353.75 " 

1 Stere = 61074.664 cubic inches. 

For the Square and Cubic Measures of other countries, take the length of the 
measure in table, page 17, and square or cube it as required. 



20 



FOREIGN WEIGHTS AND BIEASURES. 



British. 
French, 



Measures of Weight, 

1 troy Grain = .003961 cubic inches of distilled water. 
1 trov Pound = 22.815689 cubic inches of water. 



Old System.— 1 Grain 
1 Gros 
1 Once 
1 Livre 
jVew System. — Milligramme 
Centigramme 
Decigramme 
Gramme 
Decagramme 
Hecatogramme 



Spanish . 
Swedish 
Austrian . 
Prussian . 



0.8188 grains troy. 

= 58.9548 

= 1.0780 oz. avoirdupois. 

= 1.0780 lbs. 

= .01543 troy grains. 

= .15433 

= 1.54331 

= 15.43315 

= 154.33159 

== 1543.3159 
1 Millier = 1000 Kilogrammes = 1 ton sea weight. 
1 Kilogramme . = 2.204737 lbs. avoirdupois. 
1 Pound avoirdupois — 0.4535685 Kilogramme. 
1 Pound troy . = 0.3732223 " 

1 " . = 1.0152 lbs. avoirdupois. 

1 " . =0.9376 

1 " . =1.2351 

1 » . =1.0333 



Note.— /n the new French system, the values of the base of each measure, viz^ 
Metre, Litre, Stere, Arc, and Gramme, are decreased or increased by the following 
words prefixed to them. Thus, 



Milli expresses the 1000th part. 

Centi " 100th " 

Deci " 10th " 

Deca '* 10 times the value. 



Hecato expresses 100 times the value. 
Chilio " 1000 

Myrio " 10000 



Table showing the relative value oi Foreign Weights compared with 

British. 







Number 








Nnmber 






equal to 






equal to 


Places. 


Weights. 


lOU avoir- 
dupois 
pounds. 


Plaxxi. 


._[ Weights. 


lOO avoir- 
dupois 
pounds. 


Aleppo . . . 


Rottoli 


20.46 


Hanover . . 


Pound 


93.20 


Oke 


35.80 


Japan . . 




Catty 


76.92 


Alexandria 


Rottoli 


107. 


Leghorn . 




Pound 


133.56 


Algiers . . . 
Amsterdam . 




84. 


Leipsic . . 




"■ (common) 


97.14 


Pound 


91.8 


Lyons . 




" (silk) 


98.81 


Antwerp . . 
Barcelona . . 


" 


96.75 


Madeira 




" 


143.20 


u 


112.6 


Mocha . 




Maund 


33.33 


Batavia . . . 


Catty 


76.78 


Morea . 




Pound 


90.79 


Bengal . . . 
Berlin . . . 


Seer 


53.57 


Naples . 




Rottoli 


50.91 


Pound 


96.8 


Rome . 




Pound 


133.69 


Bologna . . . 
Bremen . . . 




125.3 


Rotterdam 




" 


91.80 


u 


90.93 


Russia . 




" 


110.86 


Brunswick . . 


" 


97.14 


Sicily . 




" 


142.85 


Cairo .... 


Rottoli 


105. 


Smyrna 




Oke 


36.51 


Candia . . . 




85.9 


Sumatra 




Catty 


35.56 


China . . . 


Catty 


75.45 


Sweden 




Pound 


106.67 


Constantinople 


Oke 


35.55 


" 




" 


120.68 


Copenhagen . 
Corsica . . . 


Pound 


90.80 


Tangiers 




" (miner's) 


94.27 




131.72 


Tripoli . 




Rottoli 


89.28 


Cyprus . . . 
Damascus . . 


Rottoli 


19.07 


Tunis . 




" 


90.09 




25.28 


Venice . 




Pound (heavy) 


94.74 


Florence . . 


Pound 


133.56 


" 




" (light) 


150. 


Geneva . . . 


" (heavy) 


82.35 


Vienna . 




(i 


81. 


Genoa . . . 


:; :; 


92.86 


Warsaw 




4< 


112.25 


Hamburgh . . 




93.63 









SCRIPTURE AND ANCIEJMT MEASURES. 



21 



Scripture Long Measures. 



A digit . 
A palm . 
A span 


Feet. Inches. 
. =0 0.912 
. =0 3.648 
. =0 10.944 


A cubit . • 
A fathom . 


Feet. Inches. 
. =1 9.888 
. =7 3.552 




Grecian Long Measures. 




A digit . 
A pous (foot) 
A cubit . 


Feet. Inches. 
. =0 0.7554 
. =1 0.0875 
. =1 1.5984 


A stadium 
A mile 


Feet. Inches. 
. = 604 4.5 
. =4835 



A Greek or Olympic foot = 12.108 inches. 
A Pythic or natural foot = 9.768 " 

Jewish Long Measures. 



A cubit 

A Sabbath day's 
journey . 


Feet. 

, = 1.824 
. =3648. 


Feet. 
A mile . . . = 7296 
A day's journey . = 175104 
(or 33 miles 864 feet). 




Roman Long Measures. 


A digit . 
An uncia (inch) 
A pes (foot) 


Feet. Inches. 
= .72575 
= .967 
= 11.604 


Feet. Inches. 

A cubit . . = 1 5.406 
A passus . = 4 10.02 
A mile . . = 4835 




Miscellaneous. 


Arabian foot . 
Babylonian foot 


Feet. 
. = 1.095 
. = 1.140 


Feet. 

Hebrew foot . . = 1.212 
" cubit . . — 1.817 


Egyptian . 


. = 1.421 


sacred cubit = 2.002 



Note. — The above dimensions are British. 



Table 


for finding the Distance of Objects 


at Sea, in Statute Miles. 


Height in 
feet. 


Distance in 
miles. 


Height in 
feet. 


Distance in 
miles. 


Height in 
feet. 


Distance in 
miles. 


Height in 
feet. 


Distance in 
miles. 


*.582 


1. 


11 


4.39 


30 


7.25 


200. 


18.72 


1 


1.31 


12 


4.58 


35 


7.83 


300 


22.91 


2 


1.87 


13 


4.77 


40 


8.37 


400 


26.46 


3 


2.29 


14 


4.95 


45 


8.87 


500 


29.58 


4 


2.63 


15 


5.12 


50 


9.35 


1000 


32.41 


5 


2.96 


16 


5.29 


60 


10.25 


2000 


59.20 


6 


3.24 


17 


5.45 


70 


11.07 


3000 


72.50 


7 


3.49 


18 


5.61 


80 


11.83 


4000 


83.7 


8 


3.73 


19 


5.77 


90 


12.55 


5000 


93.5 


9 


3.96 


20 


5.92 


100 


13.23 


1 mile. 


96.1 


10 


4.18 


25 


6.61 


150 


16.20 







The difference in two levels is as the square of the distance. 
Thus, if the height is required for 2 miles, 

P :22 :: 6.99: 27.96 inches; 
and if for 100 miles, P : lOO^ : : 6.99 : 1.103+ miles. 
For Geographical miles, the distance for one mile is 7.962 inches. 



22 



DISTANCES. 



Example. — If a man at the foretop-gallant-mast-head of a ship, 
100 feet from the water, sees another and a large ship (hull to), how 
far are the ships apart '? 
A large ship's bulwarks are, say 20 feet from the water. 
Then, by table, 100 feet . . . . r= 13.23 
20 " . . . . = 5.92 

Distance . . 19.15 miles. 

Note. — 1-13 should be added for horizontal refraction. 

To Reduce Longitude into Time. 

Multiply the number of degrees, minutes, and seconds by 4, and 
the product is the time. 

Example. — Required the time corresponding to 50° dl\ 
oOo 31' 

4 



h.3 22' 4:' Ans. 
If time is to be reduced, then 



4)3 



22 



50 31 Ans. 

thus, 

13x15=66° 48' 15^'. 
Degrees of longitude are to each other in length, as the cosines of 
their latitudes. 



Or, multiplying by 15 ; 

h. ■m. 
4 27 



For every 5^ they are as follows 

Miles. 

60. 
59.77 



Equator 
5° 
10° 
15° 
20° 
25° 
30° 
35° 
40° 
45° 



59.09 
57.96 
56.38 
54.38 
51.96 
49.15 
45.96 
42.43 



50° 
55° 
60° 
65° 
70° 
75° 
80° 
85° 
90° 



Miles. 

38.57 
34.41 
30. 
25.36 
20.52 
15.53 
10.42 
5.23 
0.00 



VULGAR FRACTIONS. 23 



VULGAR FRACTIONS. 

A Fraction, or broken number, is one or more parts of a Unit. 

Example. — 12 inches are 1 foot. 

Here, 1 foot is the unit, and 12 inches its parts ; 3 inches^ therefore, are the one 
fourth of a foot, for 3 is the quarter or fourth of 12. 

A Vulgar Fraction is a fraction expressed by two numbers placed one above the 
other, with a line between them ; as, 50 cents is the ^ of a dollar. 

The upper number is called the J^umerator^ because it shows the number of 
parts used. 

The lower number is called the Denominator, because it denominates, or gives 
name to the fraction. 

The Terms of a fraction express both numerator and denominator ; as, 6 and 9 
are the terms of ^. 

A Proper fraction has the numerator equal to, or less than the denominator ; as, 

An Improper fraction is the reverse of a proper one ; as, ^, &c. 

A Mixed fraction is a compound of a whole number and a fraction ; as, 5|-, &c. 

A Compound fraction is the fraction of a fraction ; as, J of ^, &c. 

A Complex fraction is one that has a fraction for its numerator or denominator, or 

I 5 — 3i 

both ; as, "o, or 3^, or #, or -5, &c. 
t 4 f 6 
A Fraction denotes division, and its value is equal to the quotient obtained by di- 
viding- the numerator by the denominator ; thus, ^^ is equal to 3, and ^-^ is equal 
to ^. 



REDUCTION OF VULGAR FRACTIONS. 

To find the greatest Number that ivill divide Two or more Numbers 
without a Remainder. 

Rule.— Divide the greater number by the less; then divide the divisor by the 
remainder ; and so on, dividing always the last divisor by the last remainder, until 
nothing remains. 

Example.-— What is the greatest common measure of 1908 and 936 1 
936) 1908 (2 
1872 
36) 936 (26 

72 

216 
216 

So 36 is the greatest common measure. 

To find the least Common Multiple of Two or more Numbers, 
Rule. — Divide by any number that will divide two or more of the given numbers 

without a remainder, and set the quotients with the undivided numbers in a line 

beneath. 
Divide the second line as before, and so on, until there are no two nimibers that 

can be divided ; then the continued product of the divisors and quotients will give 

the multiple required. 

Example.— What is the least common multiple of 40, 50, and 25 ? 

5) 40 . 50 . 25 

5) 8 . 10 . 5 

2) 8. 2. 1 

4.1.1 

Then 5X5X2X4 = 200 Aria, 



24 VULGAR FRACTIONS. 

To reduce Fractions to their lowest Terms, 
Rule. — Divide the terms by any number that wi.ll divide them without a re- 
mainder, or by their greatest common measure at once. 

Example. — Reduce J|4 of ^ foot to its lowest terms. 



9Fo 



il^-lO = -9 f -^ = l%-3 = f , or 9 inches. 

To reduce a Mixed Fraction to its equivalent^ an Improper 
Fraction. 

Note. — Mixed and improper fractions are the same; thus^ 5^= y. For illus- 
tration^ see folloicing- examples : 

Rule. — Multiply the whole number by the denominator of the fraction, and to 
the product add the numerator; then set that sum above the denominator. 

Example. — Reduce 23j to a fraction. 

23x6+2 =140 
6 6 

Example. — Reduce -^-|- inches to its value in feet. 
6 
123-i-6 = 20| ; that is, 20 feet and -J or | of a foot. 

To reduce a Whole Numler to an equivalent Fraction having a 
given Denominator. 

Rule.— Multiply the whole number by the given denominator, and set the prod 
uct over the said denominator. 
Example.— Reduce 8 to a fraction vrhose denominator shall be 9. 
8X9 = 72; then '^-^ the answer. 

To reduce a Compound Fraction to an equivalent Simple one. 

Rule.— Multiply all the numerators together for a numerator, and al^the de- 
nominators together for a denominator. 

Note.— ^FAen there are terms that are common^ they may he omitted. 

Example.— Reduce § of | of § to a simple fraction. 

13 2 6 1 ^ 

2><4><3 = 24 = 4'^"^- 

1 3- "Si 1 

Or, — X— X^= 7, by cancelling the 2's and 3's. 

Example.— Reduce J of | of a pound to a simple fraction. 
sXf = 1 ^ns. 

To reduce Fractions of different Denominations to equivalent 
ones having a common Denominator. 

Rule.— Multiply each numerator by all the denominators except its own for the 
new numerators ; and multiply all the denominators together for a common de- 
nominator. 

Note. — In this, as in all other operations, whole numbers, mixed, or compound 
fractions, must first be reduced to the form of simple fractions. 

Example. — Reduce i, §, and ^ to a common denominator. 



1X3X4=12^ 

2X2X4 = 16 V =lf = If = -J:| 

3X2X3 = 18S -* -* - 



2X3X4 = 24 
The operation may be performed mentally ; thus, 
Reduce i, ^, f , and 4 ^o a common denominator. ♦ 

3 1.12 " 1__1 6_6 1— 2J> 

2 — ^- '5—'B' I—I- 2 — 1- 



VULGAR FRACTIONS. 25 

To reduce Complex Fractions to Simple ones 

«.?rrr7f^eLtbVllf^ then multiply the nu- 

Example.— Simplify the complex fraction ?f. 

2§= f 8X 5L40 5 

4f=V 3X24=^ =9 *^''"- 



ADDITION OF VULGAR FRACTIONS. 

to^yw^nVf^K^ fractions have a common denominator, add all the numerators 
together, and then place the sum over the denominators. "uiiieraiors 

reducld'^Zf'' ^mT'''^ /rflc^207i5 have not a common denominator, they must he 
reduced to one. Also, compound and complex must be reduced to simple fractimis 

Example.— Add ^ and | together. 

|-+| = |=1 Ans. 

Example.— Add ^ of ^ of ^^ to 2| of ^. 

2^off =yx|=|i. Then, ^8+11^1X31^^^^ 



SUBTRACTION OF VULGAR FRACTIONS. 

th?n''«ni;^fjtT'® ^^® fractions the same as for other operations, when necessarv- 
SmZ^dro^aTor"""^'"'"" *^^ ^^^^^' ^-^ -^ '^^ remainderovT^^e 
Example.— What is the difference between | and | ? 

Example.— Subtract | from J. 



6X9 = 54) 

3X8=:24>=:54_24_30 - ^ 



MULTIPLICATION OF VULGAR FRACTIONS. 

tor^''to£e'iherfoT«?.L^^^ previously required ; multiply all the numera- 

denoSor. numerator, and all the denominators together for a new 

Example.— What is the product of J and f 7 

3w3 9 1 /,„„ 

T^Q — 3^— 4 Ans. 

Example.— What is the product of 6 and § of 5 ? i 

Axf of 5 = ^X y = V = 20 Ans, 

c 



26 APPLICATION OF KEDrCTION OF VULGAR FRACTIONS. 



DIVISION OF VULGAR FRACTIONS. 

Rule.— Prepare the fractions as before ; then dMde the numerator by the nu- 
merator, and the denominator by the denominator, if they will exactly divide ; but 
if not, invert the terms of the divisor, and multiply the dividend by it, as in multi- 
plication. 

Example.— Divide ^-^ by |. 

To find the Value of a Fraction in Parts of the whole Number. 

Rule.— Multiply the whole number bv the numerator, and divide by the denom- 
inator ; then, if anything remains, multiply it by the parts in the next inferior de- 
nomination, and divide bv the denominator, as before, and so on as far as necessa- 
ry ; so shall the quotients placed in order be the value of the fraction required. 

Example.— What is the value of ^ of § of $9 1 

^of| = |X-^= ^-i=^^Jins. 

Example.— Reduce | of a pound to avoirdupois ounces. 
3 
1 
4) 3(0 lbs. 

16 ounces in a lb. 

4)l8_ 

12 ounces, Ans. 
Example.— Reduce ^^ of a day to hours. 

r'o X^ = ft = 7^ hours, .3/t.. 

To reduce a Fraction from one Denomination to another. 

Rule —Multiply the number of parts in the next less denominator by the nu- 
merator if the reduction is to be to a less name, but multiply by the denominator if 
to a greater. 

Example.— Reduce ^ of a dollar to the fraction of a penny. 

1 100 100 ot;^i 

Iv -V- = — r- = ^-T, the answer. 

4 "^ 1 4 1 ' 

Example.— Reduce | of an avoirdupois pound to the fraction of an ounce. 

^-X y^ = ^^ = f , the answer. 
Example. — Reduce ? of a cwt. to the fraction of a lb. 

2y^±y^23 ^ 224 __ 3_2^ ^^le answer. 
Example.— Reduce § of | of a mile to the fraction of a foot. 

^ of ^ - -«-X ^^^ - ^^-^^ = ^^^, the answer. 

3"*4~12'^l ■" 12 1' 

Example.— Reduce ^ of a square foot to the fraction of an inch. 



1^ 144 _ 144 

4^'~T~— ~T" 
For Rule of Three in Vulgar Fractions^ see page 29. 



4 -V •«'■*• 



DECIMAL FRACTIONS. 



A Decimal Fraction is that which has for its denominator a unit (1), with as 
many ciphers annexed as the numerator has places ; it is usually expressed by set- 
ting down the numerator only, with a point on the left of it. Thus, -^ is .4, y^^ 
is .85, -^-^ is .0075, and y oV¥oo ^^ •^^^^- ^^^^" ^^^'® ^^ ^ deficiency of fig- 
ures in the numerator, prefix ciphers to make up as many places as there are 
piphers in the denominator. 



DECIMAL FRACTIOA^S. 2% 

Mixed niunbers consist of a whole number and a fraction ; as, 3.25, which is the 
same as 3.f-^, or ff^. 

Ciphers on the right hand make no alteration in their value ; for .4, .40, .400 are 
decimals of the same value, each being ^, or |. 

ADDITION OF DECIMALS. • 

Rule.— Set the numbers under each other accordmg to the value of their places, 
as in whole numbers, in which state the decimal points will stand directly under 
each other. Then, beginning at the right hand, add up all the columns of numbers 
as in integers, and place the point directly below all the other points. 
Example.— Add together 25.125, 56.19, 1.875, and 293.7325. 
25.125 
56.19 

1.875 
293.7325 
376.9225 the sum. 

SUBTRACTION OF DECIMAL FRACTIONS. 

Rule. — Place the numbers under each other as in addition ; then subtract as in 
whole numbers, and point off the decimals as in the last rule. 
Example.— Subtract 15.150 from 89.1759. 
89.1759 
15.150 
^ 74!0259 Rem. 

MULTIPLICATION OF DECIMALS. 

Rule. — Place the factors, and multiply them together the same as if they were 
whole numbers ; then poi*:: off in the product just as many places of decimals as 
there are decimals in both the factors. But if there be not so many figures in the 
product, supply the deficiency by prefixing ciphers. 

Example. — Multiply 1.56 by .75. 

1.56 
.75 

780 
1092 



1.1700 Prod. 
BY CONTRACTION. 

To contract the Operation so as to retain only as many Decimal 

places in the Product as may he thought necessary. 
Rule.— Set the unit's place of the multiplier under the figure of the multipli- 
cand whose place is the same as is to be retained for the last in the product, and 
dispose of the rest of the figures in the contrary order to what they are usually 
placed in. Then, in multiplying, reject all the figures that are more to the right 
hand than each multiplying figure, and set down the products, so that their right- 
hand figures may fall in a column straight below each other ; and observe to in- 
crease the first figure in every line with what would arise from the figures omit- 
ted ; thus, add 1 for every result from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, 4 
from 35 to 44, &c., &c., and the sum of all the lines will be the product as required. 
Example.— Multiply 13.57493 by 46.20517, and retain only four places of deci- 
mals in the product. 13.574 93 

71 502.64 

54 299 72 

8 144 96+2 for 18 

27150+2 " 18 

6 79+4 " 35 

14+1 " 5 

9+2 " 21 

627.2320 



28 DECIMAL FRACTIONS. 

Example.— Multiply 27.14986 by 92.41035, and retain only five places of deci 
mals. ^ns. 2508.92806. 

DIVISION OF DECIMALS. 

Rule. — Divide as in whole numbers, and point off in tlie quotient as many places 
for decimals as the decimal places in the dividend exceed those in the divisor; but 
if there are not so many places, supply the deficiency by prefixing ciphers. 

Example. — Divide 53.00 by 6.75. 

6.75) 53.00 ( = 7.851+. 

Here 3 ciphers were annexed to carry out the division. 

BY CONTRACTION. 

Rule. — ^Take only as many figures of the divisor as will be equal to the number 
of figures, both integers and decimals, to be in the quotient, and find how many 
times they may be contained in the first figures of the dividend, as usual. 

Let each remainder be a new dividend ; and for every such dividend leave out 
one figure more on the right-hand side of the di\asor, carrying for the figures cut 
off as in Contraction of Multiplication. 

Note. — When there are not so many figures in the divisor as are required to be 
in the quotient, continue the first operation till the number of figures in the divisor be 
equal to those remaining to be found in the quotient, after ichich begin the contraction. 
Example. — Di\dde 2508.92806 by 92.41035, so as to have only four places of deci- 
mals in the quotient. 

92.410315) 2508.928106 (27.1498 
1848 207 +1 
660 721 
646 872 +2 
13 849 
9 241 
4608 
3 69 6 
912 
832+4 
80 

74 1 2 
6 
Example. — Divide 4109.2351 by 230.409, retaining only four decimals in the quo- 
tient. Ans. 17.8345. 

REDUCTION OF DECIMALS. 

To reduce a Vulgar Fraction to its equivalent Decimal. 
Rule. — Divide the numerator by the denominator, annexing ciphers to the nu- 
merator as far as necessary. 
Example.— Reduce 4 to a decimal. 

5)4^ 

.8 Ans. 

To find the Value of a Decimal in Terms of an Inferior Denomi- 
nation. 

Rule. — Multiply the decimal by the number of parts in the next lower denomi- 
nation, and cut off as many places for a remainder, to the right hand, as there are 
places in the given decimal. 

Multiply that remainder by the parts in the next lower denomination, again cut- 
ting off for a remainder, and so on through all the parts of the integer. 
Example.— What is the value of .875 dollars ? 
.875 
100 
Cents, 87,500 

2? 

Mills, 5.000 Ans. 87 cents 5 mills. 



DECIMAL FRACTIONS. 29 

Example.— What is the content of .140 cubic feet in inches ? 
.140 
1728 cubic inches in a cubic foot. 

^^•^^ ^715. 241.3-9^2^ cubic inches. 

Example.— What is the value of .00129 of a foot 7 

Ans. .01548 inches. 
Example.— What is the value of 1.075 tons in pounds 1 

Ans. 2408. 

To reduce Decimals to equivalent Decimals of higher Denomina- 
tions. 

Rule.— Divide by the number of parts in the next higher denomination, contin- 
uing the operation as far as required. 

Example.— Reduce 1 inch to the decimal of a foot. 
12 j 1.00000 

I .08333, &c., Ans. 

Example. — Reduce 14 minutes to the decimal of a day. 
601 14.00000 
24 1 .23333 

.00972, &c., Ans. 

Example.— Reduce 14" 12"' to the decimal of a minute. 
14" 12"' 
60 



852."' 
14.2" 



.23066', &c., Atis. 

Note. — When there are several numberSy to be reduced all to the decimal of the 
highest. 

Reduce them all to the lowest denomination, and proceed as for one denomi 
nation. 



.—Reduce 5 feet 10 inches and 3 barleycorns to the decimal of a yard 


Feet. Inches. Be. 


5 10 3 


12 


70 


3 


3 213. 


12 


71. 


3 


5.9166 



1.9722, &c., yards, Ans. 

RULE OF THREE IN DECIMALS. 

Rule. — Prepare the terms by reducing the vulgar fractions to decimals, com- 
pound numbers to decimals of the highest denomination, the first and third terms 
to the same name ; then proceed as in whole numbers. See Rule, page 31. 
Example.— If i a ton of iron cost ^ of a dollar, what will .625 of a ton cost? 
^=.5 I .5:. 75: -..625 

| = .75 \ .625 

.5) .46875 

.9375 dollars, Ans. 
C2 



90 DUODECIMALS. 



DUODECIMALS. 

In Duodecimals, or Cross Multiplication, the dimensions are taken in feet, inch- 
es, and twelfths of an inch. 

Rule.— Set down the dimensions to be multiplied together, one under the other, 
so that feet may stand under feet, inches under inches, &c. 

Multiply each term of the multiplicand, beginning at the lowest, by the feet in 
the multiplier, and set the result of each immediately under its corresponding 
term, carrying 1 for every 12, from one term to the other. In like manner, multi- 
ply all the multiplicand by the inches of the multiplier, and then by the twelfth 
parts, setting the result of each term one place farther to the right hand for every 
multiplier. The sum of the products is the answer. 

Example.— Multiply 1 foot 3 inches by 1 foot one inch. 

Feet. Inches. 

1 3 

1 1 

1 3 

1 3 



14 3 
Proof. — 1 foot 3 inches is 15 inches, and 1 foot 1 inch is 13 inches ; and 15X13 
= 195 square inches. Now the above product reads 1 foot 4 inches and 3 twelfths 
of an inch, and 

1 foot = 144 square inches. 
4 inches =48 " 

3 twelfths = _ 3 
195 
Example.— How many square inches are there in a board 35 feet 4^ inches long 
and 12 feet 3^ inches wide 1 



Feet. 


Inches. 


Twelfths. 




35 


4 


6 






12 


3 


4 






424 


6 









8 


10 


1 


6 






11 


9 


6 






434 3 11 

Example.— Multiply 20 feet 6^ inches by 40 feet 6 inches. 

By duodecimals, Ans. 831 feet 11 inches 3 twelfths equal 831 square feet 

and 135 square inches. 
By decimals . . 40 feet 6 inches = 40.5 

20 " 6^ " = 20.541666, &c. 
Feet . . . 831.937499 
144 



Square inches . 134.999856 

Table showing the value of Duodecimals in Square Feet, and 
Decimals of an Inch. 

Sq. feet. Sq. inches. 

1 Foot . . . . . . = 1 or 144. 

1 Inch = ^ " 12. 

1 Twelfth = j^j " 1. 

^2 of 1 twelfth = j^-g " .083333, &c. 

.jLofJ^ofdo = 20 W .006944. &c. 

Application of this Table. 
What number of square inches are there m a floor 100^ feet broad and 25 feet 6 
Inches and 6 twelfths long 1 

Ans. 2566 feet 11 mches 3 twelfths equal 2566 feet 135 inches. 



RULE OF THREE. 31 



RULE OF THREE. 

The Rule of Three teaches how to find a fourth proportional to three given 
numbers. 

It is either Direct or Inverse. 

It is Direct when more requires more, or less requires less. Thus, if 3 barrels of 
flour cost $18, what will 10 barrels cost 1 Or, if 300 lbs. of lead cost $25.50, what 
will 10 lbs. cost 1 

In both of these cases the Proportion is Direct, and the stating must be, 

As 3:18 ::10: ^ns. 60. 

300 : 25.50 : : 10 : ^ns. .85. 

It is Inverse when more requires less, or less requires more. Thus, if 6 men 
build a certain quantity of wall in 10 days, in how many days will 8 men build the 
like quantity 1 Or, if 3 men dig 100 feet of trench in 7 days, in how many days 
will 2 men perform the same work 7 

Here the Proportion is Inverse, and the stating must be. 

As 8 : 10 : : 6 : ^ns. 7^. 

2: 7::3: ^715. 10^. 

The fourth term is always found by multiplying the 2d and 3d terms together, 
and dividing the product by the 1st term. 

Of the three given numbers necessary for the stating, two of them contain the 
supposition, and the third a demand. 

Rule. — State the question by setting down in a straight line the three neces- 
sary numbers in the following manner : 

Let the 2d term be that number of supposition which is of the same denomina- 
tion as that the answer, or 4th term, is to be, making the demanding- number the 
3d term, and the other number the 1st term when the question is in Direct Propor- 
tion, but contrariwise if in Inverse Proportion, that is, let the demanding number 
be the 1st term. 

Then multiply the 2d and 3d terms together, and divide by the 1st, and the prod- 
uct will be the answer, or 4th term sought, of the same denomination as the 2d 
term. 

Note. — If the first and third terms are of different denominations, reduce them to 
the same. If, after division, there be any remainder, reduce it to the next lower de- 
nomination, and divide by the same divisor as before, and the quotient will be of this 
last denomination. 

Sometimes two or more statings are necessary, which may always be known by the 
nature of the question. 

Example 1.— K20 tons of u-on cost $225, what will 500 tons costl 

Tons. Dolls. Tons. 

20 : 225 : : 500 
500 



210) 11250i0 

5625 dollars, Ans. 
Example 2.— If 15 men raise 100 tons of iron ore in 12 days, how many men will 
raise a like quantity in 5 days 1 

Days. Men. Days. 
As 5 : 15 : : 12 
12 
5) 180 

36 men, Jlns. 
Example 3.— A wall that is to be built to the height of 36 feet was raised 9 feet 
high by 16 men in 6 days : how many men -could finish it in 4 days at the same 
rate of working ? 

Days. Men. Days. Men. 
4 : 16 : : 6 : 24 .^ns. 
Then, if 9 feet require 24 men, what will 27 feet require 1 
9 : 24 : : 27 : 72 Ans. 

Example 4.— If the third of six be three, what will the fourth of twenty be ? 

^ns. 7J. 



32 INVOLUTION ^EVOLUTION. 



COMPOUND PROPORTION. 

Compound Proportion is the rule by means of which such questions as would 
require two or more statings in simple proportion (Role of Three) can be resolved 

in one. . -, . . j j 

As the rule, however, is but little used, and not easily acquired, it is deemed 

preferable to omit it here, and to show the operation by two or more statings. 
Example.— How many men can dig a trench 135 feet long in 8 days, when 16 

men can dig 54 feet in 6 "days 1 

Feet. Men. Feet. Men. 

First . . . As 54 : 16 : : 135 : 40 

Days Men. Days. Men. 

Second . . . As 8 : 40 : : 6 : 30 ^ns. 
Example.— If a man travel 130 miles ih 3 days of twelve hours each, in how 
many days of 10 hours each would he require to travel 360 miles 1 

Miles, Davs. Miles. Days. 

First . . . As 130 : 3 : : 360 : 8.307 

Hours. Days. Hours. Days. 

Second . . . As 10 : 8.307 : : 12 : 9.9684 ^ns. 
Example.— If 12 men in 15 davs of 12 hours build a wall 30 feet long, 6 wide, 
and 3 deep, in how many days of 8 hours will 60 men build a wall 300 feet long, 8 
wide, and 6 deep 1 *^^^- ^20 days. 



INVOLUTION. 

Involution is the multiplying any number into itself a certain number of times. 
The products obtained are called Powers. The number is called the Root, or 

"^When^a number is multiplied by itself once, the product is the square of that 
number ; twice, the cube ; three times, the biquadrate, &c. Thus, of the number 5. 
5 is the Root, or 1st power. 
5X5= 25 " Square, or 2d power, and is expressed 52. 
5X5X5 = 125 " Cube, or 3d power, and is expressed 5^ 
5X5X5X5 = 625 " Biquadrate, or 4th power, and is expressed 5*. 
The little figure denoting the power is called the Index or Exponent. 
Example.— What is the cube of 9 7 -^ns. 729. 

Example.— What is the 9th power of 2 1 -^ns. 512. 

Example.— What is the cube of 1 7 -^ns. |J. 

Example.— What is the 4th power of 1.5 1 -^ns. 5.0625. 



EVOLUTION. 

Evolution is finding the Root of any number. 

The sign y/ placed before any number, indicates the square root of that number 
is required or shown. , 1. . j u •♦ 

The same character expresses any other root by placing the mdex above it. 
, , Thus, y/25 = 5, and 4+2 = ^36. 

3/ And, ^27='9vand 3/64= 4. 

Roots which only approximate are called Surd Roots. 
Rule.— Point oflf the given number from units' place into periods of two figures 

Find the greatest square in the left-hand period, and place its root in the quo- 
tient; subtract the square number from the left-hand period, and to the remainder 
bring down the next period for a dividend. , ,. . . 

Double the root already found for a divisor ; find how many times the dmsor is 
contained in the dividend, exclusive of the right-hand figure, place the result in the 
quotient, and at the right hand of the divisor. 



EVOLUTION. 33 

Multiply the divisor by the last quotient figure, and subtract the product from the 
dividend ; bring down the next period, and proceed as before. 
Note. — Mixed decimals must be pointed off both ways from units. 
Example.— What is the square root of 2 ? 

11 2.060606 (1.414, &c. 

l| 1 



100 
96 



^ll 


400 

281 


2824 11900 
4 11296 


2828 


1 604 



Example. — What is the square root of 144 1 
11 144 (12 Atis. 



22 



044 
44 



00 
Example.— What is the square root of 12 ? Ans. 3.464101. 

SQUARE ROOTS OF VULGAR FRACTIONS. 

Rule. — Reduce the fractions to their lowest terms, and that fraction to a decimal, 
and proceed as in whole numbers and decimals. 

Note. — When the terms of the fractions are squares, take the root of each and 
set one above the other ; as, | is the square root of 5-g-. 

Example.— What is the square root of ^1 Ans. 0.86602540. 

To find the 4th root of a number, extract the square root twice, and for the 8th 
root thrice, &c., &c. 

TO EXTRACT THE CUBE ROOT. 

Rule.— From the table of Roots (page 99) take the nearest cube to the given 
number, and call it the assumed cube. 

Then say, as the given number added to twice the assumed cube is to the assu- 
med cube added to twice the given number, so is the root of the assumed cube to 
the required root, nearly. 

And, by using in like manner the root thus found as an assumed cube, and pro- 
ceeding as above, another root will be found still nearer , and in the same manner 
as far as may be deemed necessary. 

Example.— What is the cube root of 10517.9 ? 
Nearest cube, page 99 , 10648, root 22. 
10648. 10517.9 



21296 21035.8 
10517.9 10648. 



31813.9 : 31683.8 : : 22 : 21.9+ Ans. 

To extract any Root whatever. 

Let P represent the number, 

n " the index of the power, 
A " the assumed power, r its root, 
R " the required root of P. 
Then say, as the sum of w+lxA and n— IXP is to the sum of n+lXP and 
n — IXA, so is the assumed root r to the required root R. 
Example.— What is the cube root of 1500 1 
The nearest cube, page 99, is 1331, root 11. 



34 ARITHMETICAL PROGRESSION. 

P = 1500, 71 = 3, A = 1331, r=ll; 



Uien, 



n+lXA = 53^, Ti+lXP =6000 
n— IXP = 3000, 7i— IX A = 2662 

8324 : 8662 : : 11 : 11.446+ Ans, 



ARITHMETICAL PROGRESSION. 

Arithmetical Progression is a series of numbers increasing or decreasing by 
a constant number or diiference; as, 1, 3, 5, 7, 9, 15, 12, 9, 6, 3. Tiie numbers 
which form the series are called Terms; the first and last are called the Ex- 
tr ernes, and the others the Means. 

When any three of the following parts are given, the remaining two can be 
found, viz. : The First term, the Last term, the JSTumber of terms, the Common 
Difference, and the Sum of all the terms. 

When the First Term, the Common Difference, and the Number 
of Terms are given, to find the Last Term. 
Rule.— Multiply the number of terms less one, by the common difference, and to 
the product add the first term. 

Example.— A man travelled for 12 days, going 3 miles the first day, 8 the second, 
and so on ; how far did he travel the last day 7 

12—1x5-1-3 = 58 Ans. 

When the Number of Terms and the Extremes are given, to find 
the Common Difference. 
Rule.— Di\ide the difference of the extremes, by one less than the ntimber of 
terms. 

Example.— The extremes are 3 and 15, and the number of terms 7 ; what is the 
ccsnmon difference % 

15— 3-i-(7— 1) = 2 Ans. 

When the Extremes and Number of Temis are given, to find the 

Sum of all the Terms. 

Rule.— Multiply the number of terms by half the sum of the extremes. 

Example.— How many times does the hammer of a clock strike in 12 hours 1 

12X(13-r-2)=78 Ans. 

When the Common Difference and the Extremes are given, to find 
the Number of Terms. 
Rule.— Divide the difference of the extremes by the common difference, and add 
one to the quotient. 

Example.— A man travelled 3 miles the first day, 5 the second, 7 the third, and 
so on, till he went 57 miles in one day. How many days had he travelled at the 
close of the last day 1 

57— 3-r-2+l = 28 Ans. 

To find two Arithmetical Means betiveen two given Extremes. 

Rule.— Subtract the less extreme from the greater, and divide the difference by 
3, and the quotient will be the common difference, which, being added to the less 
extreme, or taken from the greater, will give the means. 
Example.— Find two arithmetical means between 4 and 16. 
]fi — 1-7-3= 4 com. dif. 
4-|-4 = 8 one mean. 
16 — i = 12 second mean. 

To find any Number of Arithmetical Means between two Extremes, 
Rule.— Subtract the less extreme from the greater, and divide the difference by 
one more than the number of means required to be found, and then proceed as in 
the foregoing rule. 



GEOMETRICAL PROGRESSION. 



35 



GEOMETRICAL PROGRESSION. 

Geometrical Progression is any series of numbers continually increasing by 
a constant multiplier, or decreasing by a constant divisor. 
As, 1, 2, 4, 8, 16, and 15, U, 3%. 

The constant multiplier or divisor is the Ratio. 

When any three of the following parts are given, the remaming two can be 
found, viz. : The First term, the Last term, the Number of terms, the Ratio, 
and the Sum of all the Terms. 

When the Ratio, Number of Terms, and the First Term are given, 
to find the Last Term. 

Rule.— Write a few of the leading terms of the series, and place their indices 
over them, beginning with a cipher. Add together the most convenient indices, to 
make an index less by one than the number of the term sought. 

Multiply together the terms of the series or powers belonging to those indices, 
and the product, multiplied by the first term, will be the answer. 

Note.— W^/tcTi the first term is equal to the ratio, the indices must begin with a 
unit. 

Example.— The first term is 1, the ratio 2, and the number of terms 23 ; what is 
the last term 1 

Indices. 01234 5 6 7 
Terms. 1, 2, 4, 8, 16, 32, 64, 128. 
1^2+3+4+54-7 = 22. 

128X32X16X8X4X2 = 4194304X1 = 4194304 Ans. 
Example.— If one cent had been put out at interest in 1630, what would it have 
amounted to in 1834 if it had doubled itself every 12 years 1 
1834—1630 = 204-i-12 = 17. 
12 3 4 5 6 

1, 2, 4, 8, 16, 32, 64, 1+2+3+4+6 = 16. 
1X2X4X8X16X64 = 65538X2 = $1,310.72 Ans 

When the First Term, the Ratio, and the Number of Teivns are 
given, to find the Sum of the Series. 

Rule.— Raise the ratio to a power whose index is equal to the number of terms, 
from which subtract 1 ; then divide the remainder by the ratio less 1, and multiply 
the quotient by the first term. 

Example.— If a man were to buy 12 horses, giving 2 cents for the first horse, 6 
cents for the second, and so on, what would they cost him 1 

312 = 531441—1 = 531440-r.(3— 1) = 2 = 265720X2 = $5,314.40 Ans. 

By another Method, the greater Extreme being known. 
(Greater extreme X ratio) -less extreme ^ g^^ ^^ ^^^ g^^.^^^ 
Ratio —1 
354294x3—2 = 1062880 



Thus 



3—1 



5.314.40, ^715., as above. 



A TABLE OF GEOMETRICAL PROGRESSION, 

Whereby any questions of Geometrical Progression proceeding from 1, and 
of double ratio J may he solved by inspection, if the number of terms ex- 
ceed 7wt 50. 



1 


8 


128 


15 1 


2 


9 


256 


16 


4 


10 


512 


17 


8 


11 


1024 


18 


16 


12 


204S 


19 


32 


13 


4096 


20 


64 


14 


8192 


21 



16384 
32768 
65536 
131072 
262144 
524288 
1048576 



22 
23 
24 
25 
26 
27 
28 



2097152 
4194304 



16777216 
33554432 
67108864 
134217728 



36 



30 
31 
32 
33 
34 
35 
36 



PERMUTATION- 


-COMBINATION — 


-POSITION. 




Table— (Continued.) 






268435456 


37 


68719476736 


44 


8796093022208 


536870912 


38 


137438953472 


45 


17592186044416 


1073741824 


39 


274877906944 


46 


35184372088832 


2147483648 


40 


549755813888 


47 


70368744177664 


4294967296 


41 


1099511627776 


48 


140737488355328 


8589934592 


42 


2199023255552 


49 


281474976710656 


17179869184 


43 


4398046511104 


50 


562949953421312 


34359738368 


100 i 633825300114114700748351602688 



PERMUTATION. 

Permutation is a rule for finding how many different ways, any given number 
of things may be varied in their^ position. 

Rule.— Multiply all the terras continually together, and the last product will be 
the answer. 

Example. — How many variations will the nine digits admit of 7 
1X2X3X4X5X6X7X8X9 = 362880 .ans. 



COMBINATION. 

Combination is a rule for finding how often a less number of things, can be 
chosen from a greater. 

Rule.— Multiply together the natural series, 1, 2, 3, &c., up to the number to be 
taken at a time. Take a series of as many terms, decreasing by 1, from the nuna- 
ber out of which the choice is to be made, and find their continued product. Di- 
vide this last product by the former, and the quotient is the answer. 
Example. — How many combinations may be made of 7 letters out of 12? 
IX 2X 3X4X5X6X7 := 5040. 
12X11X10X9X8X7X6 = 3991680-J-5040izi: 792 Ans. 
Example. — How many combinations can be madje of 5 letters out of 10? 
10X9X8X7X6 ^^^ ^ 
— — - — - — - — r=2o2 Ans. 
1X2X3X4XO 



POSITION. 

Position is of two kinds. Single and Double, and is determined by the number 
of Suppositions. 

SINGLE POSITION. 

Rule.— Take any number, and proceed with it as though it were the correct 
one ; then say, as the result is to the given sum, so is the supposed number to the 
number required. 

Example.— A commander of a vessel, after sending away in boats i, |, and ^ of 
his crew, had left 300 ; what number had he in conunand 1 
Suppose he had . 600. 
^ of 600 is 200 
|of600isl00 
I of 600 is 150 450 

150 : 300 : : 600 : 1200 Ans. 
Example.— A person being asked his age, replied, iff of my age be multiplied by 
2, and that product added to half the years I have lived, the sum will be 75. How 
old was he ? -^w^- 37^ years. 

DOUBLE POSITION. 
Rule.— Take any two numbers, and proceed with each according to the condi- 



FELLOWSHIP DOUBLE FELLOWSHIP. 



37 



tions of the question ; multiply the results or errors by the contrary supposition ; 
that is, the first position by the last error, and the last position by the first error. 

If the errors be too great, mark them + ; and if too little, — . 

Then, if the errors are alike, divide the difference of the products by the differ- 
ence of the errors ; but if they are unlike, divide the sum of the products by the sum 
of the errors. 

Example.— F asked G how much his boat cost; he replied that if it cost him 6 



mes as much as it did, and $30 more, it would stand him in S300. What Was the 


ice of the boat 1 




Suppose it cost . . 60 . . 


or 30 


G times. 


6 times. 


360 


180 


and 30 more, 


and 30 more. 


390 


210 


300 


300 


90+ 


90— 


30 2d position. 


60 1st position. 


90 2700 


5400 


90 5400 




180) 8100 (45 Ans. 




720* 




900 




900 





Example. — Wliat is the length of a fish when the head is 9 inches long, the tail 
as long as its head and half its body, and the body as long as both the head and 
tail 1 Ans. 6 feet. 



FELLOWSHIP. 

Fellowship is a method of ascertaining gains or losses of individuals engaged in 
joint operations. 

Rule.— As the whole stock is to the whole gain or loss, so is each share to the 
gain or loss on that share. 

Example. — Two men drew a prize in a lottery', of $9,500. A paid $3, and B paid 
$2 for the ticket ; how much is each one's share "? 

5 : 9.500 : : 3 : 5.700, A's share. 
5 : 9.500 : : 2 : 3.800, B's share. 



DOUBLE FELLOWSHIP, 

Or Fellowship with Time. 

Rule.— Multiply each share by the time of its interest in the Fellowship ; then, 
as the sum of the products is to the product of each interest, so is the whole gain or 
loss to each share of the gain or loss. 

Example.— A ship's company take a prize of $10,000, which they divide accord- 
ing to their rate of pay and time of service on board. The officers have been on 
board 6 months, and the men 3 months ; the pay of the lieutenants is $100 ; mid- 
shipmen $50, and men $10 per month ; and there are 2 lieutenants, 4 midshipmen, 
and 50 men. What is each one's share 1 

2 lieutenants $100 = 200X6=1200 

4 midshipmen 50 = 200x6 = 1200 

somen 10 = 500x3 = 1500 



Lieutenants 
Midshipmen 
Men . 



3900 : 1200 
3900 : 1200 
3900 : 1500 



3900 
10.000 : 3.076.92-r- 2 = $1,538.46 
10.000 : 3.076.92-^ 4 = $769.23 
10.000 : 3.846.16-J-50= $76.92 



38 ALLIGATION COMPOUND INTEREST. 



ALLIGATION. 

Alligation is a method of finding the mean rate or quality of different materials 

when mixed together. 

When it is required to find the mean price of the mixture, observe the following 
Rule. — Multiply each quantity by its rate, then divide the sum of these products 

by the sum of the quantities, and the quotient will be the rate of the composition. 

Example. — If 10 lbs. of copper at 20 cents per lb., 1 lb. of tin at 5 cents, and 1 lb 
of lead at 4 cents, be mixed together, what is the value of the composition 1 
10X20 I3Z 200 
IX 5= 5 
_1X 4= 4 

12 ) 209 (17.3^ Ans. 

When the Prices and Mean Price are given, to find lohat Quantity 
of mch Article must be taken. 

Rule 1. — Connect with a line each price that is less than the mean rate with 
one or more that is greater. 

Write the difference between the mixture rate and that of each of the simples 
opposite the price with which it is connected ; then the sum of the differences 
against any price will express the quantity to be taken of that price. 

Example. — How much gunpowder, at 72, 54, and 48 cents per pound, will com- 
pose a mixtiue worth 60 cents a pound 7 

( 48 \ 12, at 48 cents ) 

m{5\\) 12, at 54 cents V Ans. 

1 12 J 12+5 = 18, at 72 cents ) 



Proof.— 12X48+12X54+18X72 = 2520-r-12+12+12+6 = 60. 

Should it be required to mix a definite quantity of any one article, the quantities of 
each, determined by the above rule, must be increased or decreased in the proportion 
they bear to the defined quantity. 

Thus, had it been required to mix 18 pounds at 48 cents, the result would be 18 
at 48, 18 at 54, and 27 at 72 cents per pound. 

Again, when the whole composition is limited, say, 

As the sum of the relative quantities, as found by the above rule, is to the whole 
quantity required, so is each quantity so found to the required quantity of each. 

Example. — Were 100 pounds of the above mLxture wanting, the result would be 
obtained thus : 

As 42 : 100 : : 12 : 28 f. 
42 : 100 : : 12 : 284. 
42 : 100 : : 18 : 421. 



COMPOUND INTEREST. 

If any principal be multiplied by the amount (in the following table) opposite 
the years, and under the rate per cent., the sum will be the amount of that princi-' 
pal at compound interest for the time and rate taken. 

Example.— What is the amount of $500 for 10 years, at 6 per cent. ? 
Tabular number . 1.79084X500 = $895.42 Ans. 



DISCOUNT EQUATION OF PAYMENTS. 



39 



Table showing tlie amount of £\ or%\^ c^c.^for any number of years 
not exceeding 24, at the rates of 5 and 6 per cent, compound interest. 



Years. 


5 per cent. 


1 


1.05 


2 


1.1025 


3 


1.15762 


4 


1.21550 


5 


1.27G28 


6 


1.34009 


7 


1.40710 


8 


1.47745 


9 


1.55152 


10 


1.62889 


11 


1.71033 


12 


1.79585 



6 per cent. 


Years. 


5 per cent. 


6 per cent. 


1.06 


13 


1.88564 


2.13292 


1.1236 


14 


1.97993 


2.26090 


1.19101 


15 


2.07892 


2.39655 


1.26247 


16 


2.18287 


2.54035 


1.33322 


17 


2.29201 


2.69277 


1.41851 


18 


2.40661 


2.85433 


1.50363 


19 


2.52695 


3.02559 


1.59384 


20 


2.65329 


3.20713 


1.68947 


21 


2.78596 


3.39956 


1.79084 


22 


2.92526 


3.60353 


1.89829 


23 


3.07152 


3.81974 


2.01219 


24 


3.22509 


4.04893 



DISCOUNT. 

The Time, Rate per Cent., and Interest being given, to find the 
Principal. 

Rule.— Divide the given interest by the interest of $1, for the given rate and 
time. 

Example.— What sum of money at 6 p§r cent, will in 14 months gain ^14 1 
As .07-i-$14 = $200 Ans. 

The Principal, Interest, and Time being given, to find the Rate 
per Cent. 

Rule.— Divide the given interest by the interest of ihe given sum, for the time, 
at 1 per cent. 

Example.— A broker received $32.66 interest for the use of $400, 14 months; 
what was that per cent. ? 
The interest on $32.66 for 14 months, is 4.66. 

Then, as 4.66H-32.66 = 7 per cent., Ans. 

The Principal, Rate per Cent., and Interest being given, to find 
the Time. 

Rule.— Divide the given interest by the interest of the sum at the rate per cent, 
for one year. 

Example.— In what time will $108 gain 11.34, at 7 per cent.1 
The interest on $108 for one year is 7.56. 

Then, as 7.56-j-11.34 = 1.5 years, Ans. 



EQUATION OF PAYMENTS. 

Multiply each sum by its time of paynnent in days, and divide the sum of the 
products by the sum of the payments. 

Example.— A owes B $300 in 15 days, $60 in 12 days, and $350 in 20 days ; when 
is the whole due "? 

300X15 = 4500 
60X12= 720 
350X20 = 7000 
710 ) 12220 (17+ days, Ans. 



40 



ANNUITIES. 



ANNUITIES. 

The Annuity, Time, and Rate of Interest given, to find the 
Amount. 

Rule.— Raise the ratio to a power denoted by the time, from which subtract 1 ; 
divide the remainder by the ratio less 1, and the quotient, multipUed by the annui- 
ty, will give the amount. 

Note. — $1 or £1 added to the given rate per cent, is the ratio, and the preceding- 
table in Compound Interest is a table of ratios. 

Example.— What is the amount of an annual pension of $100, interest 5 per 
cent., which has remained unpaid for four vears? 

1.05 ratio ; then 1.05^ — 1.215506:25—1 — .:21o50625-i-(1.0^—l).05= 4.310125X100 = 
431.0125 dollars. 

The Annuity, Time, and Rate given, to find the Present Worth. 

Rule.— Divide the annuity by the ratio involved to the time, gubtraot-tbe'^fto- 
.tien U r om th e ana uiui^ and the remainder will be the present worth. 

Example.— What is the present worth of a pension or salary of $500, to continue 
10 years at 6 per cent, compound interest ] 

§500, by the last rule, is worth $0590.3975, which, divided by 1.0610 (bv table 

page 39, is 1.79084) = $36c0.05 ^ns. ' 

Or, by the following table, multiply the tabular number by the given annuity 

and the product will be the present worth : * 

Table showing the present worth of ^1 or £\ anmiitij, at 5 and 6 per 
cent, compound interest Jor any nuniber of years under 34. 

Years. 5 per cent. 6 per cent. Years. 5 per cent. , 6 per cent. 



1 


0.95238 


0.94339 


18 


11.68958 


10.8276 


2 


1.85941 


1.83339 


19 


12.08.532 


11.15811 


3 


2.72325 


2.67301 


20 


12.46221 


11.46992 


4 


3.54595 


3.4651 


21 


12.82115 


11.76407 


5 


4.32948 


4.21236 


22 


13.163 


12.04158 


6 


5.07569 


4.91732 


23 


13.48807 


12.30338 


7 


5.78637 


5.58238 


24 


13.79864 


12.5.5035 


8 


6.46321 


6.20979 


25 


14.09394 


12.78335 


9 


7.10782 


6.80169 


26 


14.37518 


13.00316 


10 


7.72173 


7.36008 


27 


14.64303 


13.21053 


11 


8.30641 


7.88687 


28 


14.89813 


13.40616 


12 


8.86325 


8.38384 


29 


15.14107 


33.59072 


13 


9.39357 


8.85268 


30 


15.37245 


13.76483 


14 


9.89864 


9.29498 


31 


15.59281 


13.92908 


15 


10.37966 


9.71225 


32 


15.80268 


14.08398 


16 


10.83777 


10.10589 


33 


36.00255 


14.22917 


17 


10.27407 


10.47726 


34 


16.1929 


14.36613 



Example. — Same as above ; 10 years at 6 per cent, gives 
7.30008X500 =: $3680.04 Jlns. 
^ When annuities do not commence till a certain period of time, they are said to be 
in Reversion. 

To find the Present Worth of an Annuity in Reversion. 
Rule.— Take two numbers under the rate in the above table, viz., that oppo- 
site the sum of the two given times and that of the time of reversion, and multiply 
their difference by the annuity, and the product is the present worth. 

Example.— What is the present worth of a reversion of a lease of $40 per an- 
num, to continue for six years, but not to commence until tlie end of 2 years, al- 
lowing 6 j)er cent, to the purchaser 7 

By table, 8 years .... 6.20979 
" 2 " .... 1.83339 

4.37640X40= $175.05 ^7w< 



PERPETUITIES CHRONOLOGICAL PROBLEMS . 



41 



For half yearly and quarterly payments, the amount for the given time, multi- 
plied by the number in the following table, will be the true amount : 

Rate per ct. Half yearly. Quarterly. 



Rate per ct. 



3 

3^ 
4 

4i 
5 



Half yearly. 



1.007445 
1.008675 
1.009902 
1.011126 
1.012348 



Quarterly. 



1.011181 
1.013031 
1.014877 
1.016720 
1.018559 



5i 
6 

7 



1.013567 
1.014781 
1.015993 
1.017204 



1.020395 
1.022257 
1.024055 
1.025880 



Example.— What will an annuity of $50, payable yearly, amount to in 4 years 
at 5 per cent., and what if payable half yearly 1 

By table, page 39, 
1.21550— 1~(1.05—1) =4.310X50 = $2J5.50 Jlns., for yearly payment, 
and . . 215.50X1.012348 = $218.16 " half yearly do. 



PEEPETUITIES. 

Perpetuities are such annuities as continue forever. 

Rule.— Divide the annuity by the rate per cent., multiply by the tabular num- 
ber above, and the quotient will be the answer. 

Example.— What is the present worth of a $100 annuity, payable semi-annually, 
at 5 per cent. 1 

100-r.05 = 2000X1.012348 (from preceding table) = $2,024.70 ^ns. 
For Perpetuities in Reversion, subtract the present worth of the annuity for the 
time of reversion from the worth of the annuity, to commence immediately. 

Example.— What is the present worth of an estate of $50 per annum, at 5 per 
cent., to commence in 4 years 1 

50-f-.05 = 1000. 

$50, for 4 years, at 5 per cent. = 3.54595 (from table) X50 = 177.29 

$822.71 ^ns.y 
which in 4 years, at 5 per cent, compound interest, would produce $1000. 



CHRONOLOGICAL PROBLEMS. 

The Golden Number is a period of 19 years, in which the changes of the moon 
fell on the same days of the month as before. ' 

To find the Golden Number, or Lunar Cycle. 

Rule. — Add one to the given year ; divide the sum by 19, and the remainder is 
the golden number. 
Note. — IfQ remain^ it will be 19. 
Example.— What is the golden number for 1830 1 

1830+1-M9 = 96 rem. : 7 Ans. 

To find the Epact. 

Rule. — Divide the centuries of the given year by 4 ; multiply the remainder by 
17, and to this product add the quotient, multiplied by 43; divide this sum plus 86 
by 25, multiplying the golden number by 11, from wliich subtract the last quotient, 
and, rejecting the 30's, the remainder will be the answer. 

Example. — Required the epact for 1830. 
Centuries. 18-M = 4|. 2X17 = 34. 4X43= 172+34 = 206+86 = 292-^25 = 11, 

last quotient. 
Golden number, as ascertained above, 7X11 =: 77 — 11 (last quotient) = 66, rejecting 
30's=6 Ans. 
Example.— What is the epact for 1839 1 Ans. 15. 

D2 



42 



TABLE OF EPACTS, DOMINICAL LETTERS, ETC. 



TO FIND THE MOOX'S AGE ON ANY GIVEN DAY. 
Rule.— To the day of the month add the epact and number of the month, then 
reject the 30's, and the answer will be the moon's age. 



January 0, 
February'- 2, 
March 1, 



Example.— For 5th February, 1841. 
Given day 
Epact 
Number of month . 



Numbers of the Month. 

April 2, I July 5, 

May 3, August 6, 

June 4, I Septembers, 



October 8, 
November 10, 
December 10. 



¥ 



age of the moon. 



The Cycle of the Sun is the 28 years before the days of the week return to the 
same days of the month. 



Table of Epacts^ Dominical Letters ^ and an Almanac^ from 1776 to 1875. 



February, 

March, 

November, 


February,* 
August. 


May. 


January. 
October. 


January,* 

April, 

July. 


September, 
December. 


June. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


16 


17 


18 


19 


20 


21 


22 


33 


24 


25 


2o 


27 


28 


29 


30 


31 











N. B. — In leap-: 
marked *. 


-ear, 


January 


and 


February must be taken in 


the columns 


Years 


Days. 


Dom. 
Let- 
ters. 


*: 
% 

a 


Years 


IDom. 

Days. Let- 

1 ters. 


% 


Years 


Dom. 
Days. Let- 
ters. 


§ 

a 


Years Days 


Dom. 
Let- 
ters. 


1 


1776 


Friday* 


GF 


9 


1801 [Sunday. 


1 D 


15 


1826 1 Wedn'y. 


A 


22 


1851iSat'y. 


E 


28 


1777 


Saturd'y 


E 


20 


1802 Monday. 


C 


26 


1827 Thursd. 


G 


3 


1852 Mon.* 


DC 


9 


1778 


Sunday. 


D 


1 


1803 


Tuesd'y. 


B 


7 


1828Saturd.* 


FE 


14 


1853, Tues. 


B 


20 


1779 


Monday. 


C 


12 


1804 


Thurs.* 


AG 


18 


1829 


Sunday. 


D 


25 


1854 Wedn. 


A 


] 


1780 


Wedn.* 


BA 


23 


1805 


Friday. 


F 


29 


1830 


Monday. 


C 


6 


1855iThur. 


G 


12 


1781 


Thiirsd. 


G 


4 


1806 


Saturd'y 


E 


11 


1831 


Tuesd'y. 


B 


17 


1856 Sat' y* 


FE 


23 


1782 


JFriday. 


F 


15 


1807 


Sunday. 


D 


22 


1832 Thurs.* 


AG 


28 


1857 Sund. 


D 


4 


1783 


Saturd'y 


E 


26 


1808 


Tuesd.* 


CB 


3 


1833 Friday 


F 


9 


1858 Mond. 


C 


15 


1784 


Mond.* 


DC 


7 


1809 


Wedn'y. 


A 


14 


1834 Saturd'y 


E 


20 


1859 Tues. 


B 


26 


1785 


Tuesd'y 


B 


18 


1810 


Thursd. 


G 


25 


1835; Sunday. 


D 


1 


1860 iThu.* 


AG 


7 


1786 


Wedn'y. 


A 


29 


1811 


Friday. 


F 


6 


1836 Tuesd * 


CB 


12 


1861 ! Friday 


F 


18 


1787 


Thursd. 


G 


11 


1812 


Sunday* 


ED 


17 


1837j Wedn'y. 


A 


23 


1862 


Satur. 


E 


29 


1788 


Saturd.* 


FE 


22 


1813 


Monday. 


C 


28 


1838; Thursd. 


G 


4 


1863 


Sund. 


D 


11 


1789 


Sunday. 


D 


3 


1814 


Tuesd'y. 


B 


9 


1839 Friday. 


F 


15 


1864 |Tue.* 


CB 


22 


1790 


Monday. 


C 


14 


1815 


Wedn'y. 


A 


20 


1840Sund'y.* 


ED 


26 


1865 


Wedn.' 


A 


3 


1791 


Tuesd'y. 


B 


25 


1816 


Friday.* 


GF 


1 


1841 Monday. 


C 


7 


1866 


Thur. 


G 


14 


1792 


Thurs.* 


AG 


6 


1817 


Saturd'y 


E 


12 


1842|Tuesd'y. 


B 


18 


1867 


Friday 


F 


25 


1793 


Friday.' 


F 


17 


1818 


Simday. 


D 


23 


1843 Wedn'y. 


A 


29 


1868 


Sun.* 


ED 


6 


1794 


Saturd'y 


E 


28 


1819 


Monday. 


C 


4 


1844 j Friday.* 


GF 


11 


1869 


Mond. 


C 


17 


1795 


Sunday. 


D 


9 


1820 


Wedn.* 


BA 


15 


1845 Saturd'y 


E 


22 


1870 


Tues 


B 


28 


1796 


Tuesd.* 


CB 


20 


1821 


Thursd. 


G 


26 


1846 Sunday. 


D 


3 


1871 


Wedn. 


A 


9 


1797 


Wedn'y. 


A 


1 


1822 


Friday. 


F 


7 


1847 Monday. 


C 


14 


1872 


Frid.* 


GF 


20 


1798 


Thursd. 


G 


12 


1823 


Saturd'y 


E 


18 


1848 Wed'y.* 


BA 


25 


1873 


Satur. 


E 


1 


1799 


Friday. 


F 


23 


18$M 


Mond'y* 


DC 


29 


184»Thursd. 


G 


6 


1874 


Sund. 


D 


12 


1800 


Saturd'y 


E 


4 


1825 


Tuesd'y. 


B 


11 


1850 Friday. 


F 


17 


1875 


Mond. 


C 


23 



* Distinguishes the leap-years. 



PROMISCUOUS QUESTIONS. 43 

Use of the above Table— To find the day of the week on which any given day of 
the month falls in any year from 1776 to 1875. 

Example.— The great fire occurred in New- York on the 16th December, 1835; 
what was the day of the week 1 
Against 1835 we find Sunday, and at top, under December, we find that the 13th 
^ was Sunday ; consequently, the 16th was Wednesday. 



PEOMISCUOUS QUESTIONS. 

1. If SlOO principal gain $5 interest in one year, what amount 
will gain $20 in 8 months 1 

As 12 months : 5 : : 8 months : 3.33, the interest for 8 months. 
And, as 3.33 : : 100 : : 20 : 600 the answer. 

2. A reservoir has two cocks, through which it is supplied ; by 
one of them it will fill in 40 minutes, and by the other in 50 min- 
utes ; it has also a discharging cock, by which, when fall, it may be 
emptied in 25 minutes. If the three cocks are left open, in what 
time would the cistern be filled, assuming the velocity of the water 
to be uniform ] 

The least common multiple of 40, 50, and 25 is 200. 

Then . . the 1st cock will fill it 5 times in 200 minutes, 
the 2d *' 4 " 200 

or both 9 times in 200 minutes ; and, as the discharge-cock will 
empty it 8 times in 200 minutes, then 9—8 = 1, or once in 3.20 
hours, Ans. 

3. Out of a pipe of wine, containing 84 gallons, 10 gallons were 
drawn off, and the vessel replenished with water ; after which 10 
gallons of the mixture was likewise drawn off, and then 10 gallons 
more of water were poured in, and so on for a third and fourth time. 
It is required to find how much pure wine remained in the vessel, 
supposing the two fluids to have been thoroughly mixed ] 

84—10 = 74 



As 84 : 10 : 
84 : 10 : 
84 I 10 : 



74 : 8.80952 

65.19048 : 7.76077 
57.42971 : 6.83687 
6.83687 



50.59284 Ans. 



4. A traveller leaves New- York at 8 o'clock in the morning, and 
walks towards New-London at the rate of 3 miles an hour, without 
intermission ; another traveller sets out from New- London at 4 
o'clock the same evening, and walks for New-York at the rate of 4 
miles an hour, constantly ; now, supposing the distance between 
the two cities to be 130 miles, whereabout on the road will they 
meetl 

From 8 o'clock till 4 o'clock is 8 hours ; therefore, 8x3=24 
miles, performed by A before B set out from New-London ; and, con- 
sequently, 130—24 = 106 are the miles to travel between them after 



44 PROMISCUOUS QUESTIONS. 

that. Hence, as 7 = 3+4 : 3 : : 106 : -^^^ 45f more miles trav- 
elled by A at the meeting ; consequently, 24+45^ z=z 69^ miles from 
New-York is the place of their meeting. 

5. What part of ^3 is a third part of $2 1 

loff ofl = lxfxJ = f Ans. 

6. The hour and minute hand of a clock are exactly together at 
12 ; when are they next together ] 

As the minute hand runs 11 times as fast as the hour hand ; then, 
11 ; 60 : : 1 : 5 mm. 5^ sec. The time, then, is 5 min. 5^^ sec. 
past 1 o'clock. 

7. The time of the day is betw^een 4 and 5, and the hour and min- 
ute hands are exactly together ; what is the time ? 

The speed of the hands is as 1 to 11. 

4 hours X60 =240, which -f-11 = 21^^ min. added to 4 hours, 
Ans. 

8. A can do a piece of work in 3 weeks, B can do thrice as 
much in 8 weeks, and C five times as much in 12 weeks ; in what 
time can they finish it jointly ] 

Week. Week. Week. 

As 3 : 1 : : 1 : 1 work done by A in one week. 
8 : 3 : : 1 : f " B 

12 : 5 : : 1 :j% " C " 

Then, by addition, ^+|4-t2 ^^'i^^ ^^ the work done by them all in 
one week ; these, reduced to a common denominator, become yj 
^^-\-^==:^=:l', whence, 9 : 6 : : 8 : 5| Ans. 

9. A cistern, containing 60 gallons of water, has 3 unequal cocks 
for discharging it ; one cock will empty it in 1 hour, a second in 2 
hours, and a third in 3 hours ; in what time will it be emptied if they 
all run together '? 

First, i w^ould run out in 1 hour by the second cock, and J by 
the third ; consequently, by the 3 was the reservoir supplied one 
hour, "l+^+l = f +f +f being reduced to a common denominator, 
the sum of these 3 = V ; whence the proportion, 1 1 : 60 : : 6 : 32 j^ 
minutes, the time required. 

10. What will a body, weighing 10 lbs. troy, lose by being carried 
to the height of 7 miles above the surface of the earth 1 

As the gravitation or iveight of a body above the earth is inversely as 
the square of its distance, and the earth's diameter being, say 3993 milesy 
then 3993+7 — 4000. 

And, as 4000^ : 3993^ : : 10 : 9.965 lbs., Ans. 

11. Suppose a cubic inch of common glass weighs 1.49 ounces 
troy, the same of sea water .59, and of brandy .53. A gallon of this 
liquor in a glass bottle, which weighs 3.84 ll^s., is thrown into the 
water. It is proposed to determine if it will sink ; and if so, how 
much force will just buoy it up 1 

3.84X12-M.49 = 30.92 cubic inches of glass in the bottle. 
231 cubic inches in a gallon x.53 = 122.43 ounces of brandy. 



PROBIISCUOUS QUESTIONS. 45' 

Then, bottle and brandy weigh 3. 84x 12+122.43 = 168.51 ounces, 
and contain 261.92 cubic inches, which, X-59 == 154.53 ounces, the 
weight of an equal bulk of salt water. 

And, 168.51 — 154.53= 13.98 ounces, the weight necessary to sup- 
port it in the water. 

12. How many fifteens can be counted with four fives 1 Ans, 4. 

13. What is the radius of a circular acre '? 

(Side of a square x 1.128 r^ diameter of an equal circle.) 
208.710321, the side of a square acre, x 1.128 =:= 235.50-1-2 (for 
radius) = 117.75 feet, Ans. 

14. From Caldwell's to Newburg is 18 miles ; the current of the 
river is such as to accelerate a boat descending, or retard one as- 
cending U miles per hour. Suppose two boats, driven uniformly at 
the rate of 15 miles per hour through the water, were to start one 
from each place at the same time, where will they meet 1 

Call X the distance from N to the place of meeting ; its distance 
from C, then, will be 18— a:. 

Speed of descending boat, 15+1.5 ~ 16.5 miles per hour. ' 
Speed of ascending boat, 15—1.5 = 13.5 miles per hour. 

— - = time of boat descending to point of meeting. 

16.5 

18 X 

= time of boat ascending to point of meeting. 

13.5 

These times are, of course, equal ; therefore, - - - = - •♦ 

Then, 13.5x = 297— 16.5a;, and 13.52;+16.5a; = 297, or 30a; = 297. 

297 
Hence, x = -— - = 9.9 miles, the distance from Newburgh, Ans. 

15. A steamboat, going at the rate of 10 miles per hour through 
the water, descends a river, the velocity of which is 4 miles per 
hour, and returns in 10 hours ; how far did she proceed] 

Let X = distance required, — — = time of going, — -— = time of returning. 

-^ - = 10 ; 6z+14x = 840 ; 20x = 840 ; 840-^-20 = 42, ^ns. 
14^6 ' ^ 

16. The flood tide wave of a river runs 20 miles per hour, the 
current of it is 3 miles per hour. Assume the air to be quiescent, 
and a floating body set in motion at the commencement of the flow 
of the tide ; how long will the body drift in one direction, the tide 
flowing six hours from each point of the river 1 

Let X be the time required ; 20x = distance the tide has run up, together with the 
distance which the floating body has moved ; 3x= whole distance which the body 
has floated. 

Then 20x— 3z = 6X20, or the length in miles of a tide. 

x= -^r— X6 = 7 hours, 3 minutes, 31^^^ seconds, ^ns. 

17. If a steamboat, going uniformly at the rate of 15 miles in an 
hour through the water, were to run for 1 hour with a current of 5 
miles per hour ; then, to return against that current ; what length of 
time would she require to reach the place from whence she started] 

15+5 = 20 miles, tbe distance gone during the hour. 

Then 15—5 = 10 miles, is her etfective velocity per hour when returning, and 
20^10 = 2 hours, the time of returning, and 2+1 = 3 hours, or the whole time 
occupied, Jins. 



46 GEOMETRY. 



GEOMETRY. 

Definitions. 

A Foint has position, but not magnitude. 

A Line is length without breadth, and is either Right, Curved, or Mixed. 
A Right Line is the shortest distance between two points. 
A Mixed Line is composed of a right and a curved line. 
A Superficies has length and breadth only, and is plane or curved. 
A Solid has length, breadth, and thickness. 

Kxi Angle is the opening of two lines having different directions, and is either Right, 
Acute, or Obtuse. 

A Right Angle is made by a line perpendicular to another, falling upon it. 
An Acute Angle is less than a right angle. 
An Obtuse Angle is greater than a right angle. 
A Triangle is a figure of three sides. 
An Equilateral Triangle has all its sides equal. 
An Isosceles Triangle has two of its sides equal 
A Scalene Triangle has all its sides unequal. 
A Right-angled Triangle has one right angle. 
An Obtuse-angled Triangle has one obtuse angle. 
An Acute-angled Triangle has all its angles acute. 

A Quadrangle or Quadrilateral is a figure of four sides, and has the following par- 
ticular names, viz. : 

A Parallelogram, having its opposite sides parallel. 

A Square, having length and breadth equal. 

A Rectangle, a parallelogram having a right angle. 

A Rhombus (or Lozenge), having equal sides, but its ang es not right angles. 

A Rhomboid, a parallelogram, its angles not being right angles. 

A Trapezium, having unequal sides. 

A Trapezoid, having only one pair of opposite sides parallel. 
Note. — A Triangle is sometimes called a Trigon, and a Square a Tetragon. 
Polygons are plane figures having more than four sides, and are either Regular 
or Irregular, according as their sides and angles are equal or unequal, and they are 
named from the number of their sides or angles. Thus : 
A Pentagon has five sides. 
'A Hexagon 



A Heptagon 
An Octagon 
A Nonagon 
A Decagon 
An Undecagon 
A Dodecagon 



SIX 

seven 

eight 

nine 

ten 

eleven 

twelve 



A Circle is a plane figure bounded by a curve line, called the Circumference (or 
"Periphery). 

An Arc is any part of the circumference of a circle. 

A Chord is a right line joining the extremities of an arc. 

A Segment of a circle is any part bounded by an arc and its chord. 

The Radius of a circle is a line drawn from the centre to the circumference. 

A Sector is any part of a circle bounded by an arc and its two radii. 
* A Semicircle is half a circle. 

A Quadrant is a quarter of a circle. 

A Zone is a part of a circle included between two parallel chords and their arcs. 

A Lune is the space between the intersecting arcs of two eccentric circles. 

A Gnomon is the space included between the lines forming two similar parallelo- 
grams, of which the smaller is inscribed within the larger, so as to have one angle in 
each common to both. 

A Secant is a line that cuts a circle, lying partly within and partly without it. 

A Cosecant is the secant of the complement of an arc. 

A Sine of an arc is a line running from one extremity of an arc perpendicular to a 
diameter passing through the other extremity, and the sine of an angle is the sine of 
the arc that measures that angle. 

The Versed Sine of an arc or angle is the part of the di'imeter intercp^^ted betwee>, 
the sine and the arc. 



GEOMETRY. 



47 



The Cosine of an arc or angle is the part of the diameter intercepted between the 
sine and the centre. 

A Tangent is a right line that touches a circle without cutting it. 

A Cotangent is the tangent of the complement of the arc. 

The Circumference of every circle is supposed to be divided into 360 equal parts 
called Degrees ; each degree into 60 Minutes, and each minute into 60 Seconds, and 
so on. 

The Complement of an angle is what remains after subtracting the angle from 90 
degrees. 

The Supplement of an angle is what remains after subtracting the angle from 180 
degrees. 

To exemplify these definitions, let A c 6, in the following diagram, be an assumed 
arc of a circle described with the radius A B. 




B k, the Cosine of the arc A c b. 

A g, the Tangent of do. 

CB b, the Complement, and 6 B E, the 
Supplement of the arc A c b. 

C g, the Cotangent of the arc, written 
coH 

B^, the Cosecant of the arc, written 
cosec. 

m b, the Coversed sine of the arc, or, by 
convention, of the angle A B 6 ; written 
coversin. 



A c &, an Arc of the circle AGED. 

A b, the Chord of that arc. 

e D rf, a Segment of the circle. 

A B, the Radius. 

A B 6 c, a Sector. 

A D E B, a Semicircle. 

C B E, a Quadrant. 

A c <i E, a Zone. 

n A, a Lune. 

E g, the Secant of the arc A c b. 

b k, the Sine of do. 

A k, the Versed Sine of do. 

A Prism is a solid of which the sides are parallelograms, and are of three, four, 
five, or more sides, and are upright or oblique. 

A Parnllelopipedon is a sohd terminated by six parallelograms : thus, a four-sided 
prism is a parallelopipedon. 

A Pyramid is a solid bounded by a number of planes, its base being a rectilinear 
figure, and its faces triangles, terminating in one point, called the summit or vertex. 

It is regular or irregular, upright or oblique, and triangular, quadrangular, and so 
on, from its equahty of sides, inclination, or number of sides. 

A Cylinder is a solid formed by the rotation of a rectangle about one of its sides, at 
rest ; this side is called the axis of the cylinder. It is right or oblique as the axis is 
perpendicular or inclined. 

An Ellipse is a section of a cylinder oblique to the axis. {See Conic Sections, 
page 54.) 

A Sphere is a solid bounded by one continued surface, every point of which is 
equally distant from a point within the sphere, called the centre. 



48 



GEOMETRY. 



The Altitude, or height of a figure, is a perpendicular let fall from its vertex to 
the opposite side, called the base. 

The Measure of an angle is an arc of a circle contained between the two lines that 
fonn the angle, and is estimated by the number of degrees in the arc. 

A Prismoid has its two ends as any unlike parallel plane figures of the same num- 
ber of sides, the upright sides being trapezoids. 

A Sphei-oid is a solid resembling the figure of a sphere, but not exactly round, one 
of its diameters being longer than the other. 

A Spindle is a solid formed by the revolution of some curve round its base. 

A Segment is a part cut off by a plane, parallel to the base. 

A Frustum is the part remaining after the segment is cut oiF. 

A Cycloid is a curve formed by a point in the circumference of a circle, revolving 
on a right line the length of that circumference. 

An Epicycloid is a curve generated by a point in one circle w^hich revolves about 
another circle, either On the concavity or convexity of its circumference. 

An Ungula is the bottom part cut off by a plane passing obhquely through the base 
of a cone or cylinder. 

The Perimeter of a figure is the sum of all its sides. 

A Problem is something proposed to be done. 

A Postulate is something required. 

A Theorem is something proposed to be demonstrated. 

A Lemma is something premised, to render what follows more easy, 

A Corollary is a truth consequent upon a preceding demonstration. 

A Scholium is a remark upon something going before it. 



. ^6c 


H . G H 










O 




• 4 


\\m\ 


















B 


i 


\\U\\\ 




















J, 




1 \m 
























(i iiiiil 
























iiiiiiiii 
























HiilHu' 






■ 


















iitHiH 
























1 1 111 1 
























\t 


lil 






















c 


m\ 


n! 




















D 



To construct a Diagonal Scale upon any Line, as A B—fig. 1. 

Di-vade the Hne into as many divisions as there are hundreds of feet, spaces of ten 
feet, feet, or inches required. 

Draw perpendiculars from every division to a parallel line C D. 

Divide these perpendiculars and one of the divisions A E, C F, into spaces of ten 
if for feet and hundredths, and into twelve if for inches ; draw the lines A 1, a 2, 6 
3, &c., and they will complete the scale required. 

Thus : The line A B representing ten feet ; A to E, E to G, &c., will measure 
one foot ; A to a, C to 1, 1 to 2, &c., will measure 1-lOth of a foot ; and the several 
lines A 1, a 2, &c., will measure upoqg;he lines k k, I I, &c., 1-1 00th of a foot ; and 
p will measure '.pon k k, 11, &c., 1-lOth of a foot. 





To circumscribe a Pentagon about a given Circle— Jig. 2. 
Rule. — Inscribe a pentagon in the circle, defining the points s r v m n. 



GEOMETRY. 



4.9 



From the centre o, draw o r, o v, &c. ^ 

Through n, m, &c.,draw A B, B C, &c., perpendicular to o n, o m, and complete 

the figure. 

Note. — Any other polygon may he made in a similar manner, by drawing tangents 

to the points, first defining them in the circle. 

Upon a given Line A B, to form an Octagon— fig. 3, 
Rule.— On the extremities of A B, erect indefinite perpendiculars A F, B E, pro- 
duce A B to m and w, and bisect the angles m K e and n B/wirh A H and B C. 
Make A H and B C equal to A B, and draw H G, C D, parallel to A F, and equal 

toAB. . . r. T, 

From G and D, as centres with a radius equal to A B, describe arcs cutting A F, B 
E, in F and E. Join G F, F E, and E D, and the figure is made. 

Circles and Squares. 





O E K A. 

To describe a Circle that shall pass through any three given Points, as XB C^fig. 4. 
Rule. — Upon the points A and B, with any opening of the dividers, describe two 
arcs to intersect each other, as at e e ; on the points A C describe two more to inter- 
sect each other in the points c c ; draw the lines e e and c c, and where these two 
lines intersect o, place one foot of the dividers, and extend the other to A, B, or C, 
and it will pass through the three given points as required. 

To make a Square equal to a given Triangle— fig. 5. 

Let B <f E be the triangle given. 

Rule. — Extend the side of the triangle B E to O, making E O equal to half the 
length of the perpendicular of the triangle A d. Divide B O into two equal parts in 
K, and with the distance K B describe the semicircle B H O. Upon E erectolhe per 
pendicular E H, which will be the side of a square, equal to the triangle B d E. 




Triangles and Squares. 






To make an Equilateral Triangle equal to two gioen Equilateral Triangles— fig. 6. 

Let the given equilateral triangles be A and B. 

Rule. — Draw a right line C D equal in length to one side of the triangle B. 
Erect the perpendicular D E, equal in length to one side of the triangle A. 

Draw C E, and complete the equilateral triangle C E F, which shall be equal io 
quantity to the two given equilateral triangles A and B. 

To make a Square equal to two given Squares— fig. 7. 
Let the two squares given be A and B. 

E 



50 



GEOMETRY. 



Rule. — Draw tlie line C F, equal in leng-th to one side of the larg'est square A 
Raise the perpendicular E F, equal in leng-th to one side of the smallest square B. 
Dra-w C E, and C E is the side of the square C E G O, which is equal in quantity 
to the two given squares A and B. 

Circles and Ellipse, 
9. 




Two Circles^ F and G, being given, to make another of equal quantity—fig. 8. 

Rule. — Upon the diameter of the largest of the two circles at the point D, erect 
the perpendicular D E, equal in length to the diameter A B of the least circle. 

Draw B E, and divide it into two equal parts m O ; take the distance B or O E, 
and describe a circle. This circle will be equal in quantity to the two given circles F 
andG. 

To describe an Ellipse of any given length, without regard to breadth— fig. 9. 

Let A B be the given length. 

Rule. — Divide it into three equal parts, as A 5 i B. Then, with the radius A 5, 
describe A F o t ti C ; and from z, the circle B D n 5 o E ; then with n F and o C de- 
scribe F E and C D, and you have the ellipse required. 



10. 



Ellipses, 



11. 




To describe an Ellipse to any length and breadth given— fig. 10. 

Let the longest diameter given be the line F, and the shortest G. 

Rule.— Make A B equal to F, and C D to G, dividing A B equally at right angles 
in a. 

Make A o equal to D C, and dividing o B into three equal parts, set off two of 
those parts from a to 6 and from a to c, then with the distance c b make the two 
equilateral triangles c db and c c b, whose angles are the centres, and the sides being 
continued are the hnes of direction for the several arcs of the oval A C B D. 

'Note.— Carpenters, Bricklayers, and Masons are oftentimes obliged to work an 
architrave, 6fC., about windows, of this form : they may, by the help of the four centres 
c, d, b, €, and the lines of direction h d, e f, d g, e i, describe another line around the 
former, and at any distance required, as h if g. 



GEOMETRY. 



51 



To describe an Ellipse to any length and breadth required^ another way— fig. 11, 

Let the longest diameter be A, and the shortest B. 

Rule.— Draw the Ime C D equal in length to A ; also E F equal in length to B, 
and at right angles with D. 

Take the distance C O or O D, and with it, from the point E and F, describe the 
arcs h arid /upon the diameter C D. 

Strike in a nail or pin at h and at/, and put a string around them, of such a length 
that the two ends may just reach to E or F. 

Introduce a pencil, and bearing upon the string, carry it around the centre O, and 
it will describe the ellipse required. 




To find the Centre and two diameters of an Ellipse— fig. 12. 

Let A B C D be the ellipse. 

Rule.— Draw at pleasure two lines, Q G, M O, parallel to each other ; bisect them 
in the points H N, and draw the line P E ; bisect it in K, and upon K, as a centre, 
describe a circle at pleasure, as F L R, cutting, the figure in the points F L ; draw the 
right line F L, bisect it in I, and through the points I K draw the greatest diameter 
A C, and through the centre K draw the least diameter B D, parallel to the line FL. 

To draw a Spiral Line about a given Point— fig. 13. 

Let B be the centre. 

Rule. — Draw A C, and divide it into twice the number of pans that there are to 
be revolutions of the line. Upon B describe K I, G F, H E, &c., and upon I describe 
K F, G E, &c. 



Polygons, 




52 



GEOMETRY. 



Upon a given line, to describe any Polygon beyond a Pentagon—Jig. 14. 

Let A B be the g-iven line. ^ ^ ^ 

Rule,— Bisect the line A B in Q, and erect the perpendicular Q P. From the 
point A describe the arc B H, and from B the arc A H, and divide B H into equal 
parts, as H 1, 2, 3, 4, 5, B. ., , . -, tt , , -,. n. 

Let a pentagon be required. From the point H, with the interval H 1, describe the 
arc I 7, and the point I will be the centre of a circle containing the given line A B 
five times, the interval I B being the radius thereof. Take the point H for the cen- 
tre of another circle, and H B for the radius ; this circle will contain the hne A B six 
times. From the point 7, with the radius 7 B, a circle drawn will contain A B seven 
times. From the point H, with the interval H 2, describe the arc 2 8 ; and from the 
point 8, with the radius 8 B, draw a circle, and A B shall be the side of an octagon. 
From 9, with the radius 9 B, you form a nine-sided figure ; from 10 a ten-sided 
figure ; and so on to 12. 

Arches. 




To describe an Elliptic Arch on the Conjugate Diameter— Jig. 15. 

Rule.— Draw the diameter A B, and in the middle at k, erect the perpendicular 
k 0, equal to the height of the arch ; divide the perpendicular k o into two equal parts 
at e ; continue the hne A B on both sides at pleasure, and from the point k, with the 
distance k o, define c and d; through c e, d e, draw c ej a.ndde g at pleasure ; d 
and c are centres for the arcs A g and B/, and e the centre for the arc g oj, which 
will form the arch required. 

To draw a Gothic Arch— Jig. 16. 

Rule 1.— Take the length of the line A B, and on the points A and B describe the 
arcs A c and B d, and it will complete the arch required. 

Rule 2, Jig. 17.— Divide the line A B into three equal parts, at c and d; take A d 
or B c, and describe B e or A c, and it will give an arch of another torm. 





Rule 2, Jig. 18.— Divide the line A B also into three equal parts, e f; tiom the 



GEOMETRY. 53 

length off B, describe the arcs A p-VnH R * „„^1 ^i '^°'"- ""^ P°'"'' « /, with the 
^ .Jnd iUni it w,n complete^fotS^r Goihli'a'^r ''' """"^ ^ "^ "'^^""^ *^ "" 

arcs . ^ ^ndfg, and it will complete another Gothic^ch"^' '^ °" ""' P""'" ' " ">« 

E2 



54 CONIC SECTIONS. 



CONIC SECTIONS. 

Definitions, 

A Cone is a solid figure having a circle for its base, and termina- 
ted in a vertex. 

Conic Sections are the figures made by a plane cutting a cone. 

An Ellipse is the section of a cone when cut by a plane obhquel 
through both sides. 

A Parabola is the section of a cone when cut by a plane paralle; 

to its side. , , , . ^ 

A Hyperbola is the section of a cone when cut by a plane, making 

a greater angle with the base than the side of the cone makes 
The Transverse Axis is the longest straight line that can be drawn 

^^ The Conjugate Axis is a line drawn through the centre, at right 
angles to the transverse axis. ^ , 

An Ordinate is a right line drawn from any point of the curve 
perpendicular to either of the diameters. 

An Abscissa is a part of any diameter contained between its ver- 
tex and an ordinate. , w *u 4. ^; 

The Parameter of any diameter is a third proportional to that di- 
ameter and its conjugate. J. . • 14-^ 

The Focus is the point in the axis where the ordinate is equal to 
half the parameter. u i^ ^,. 

A ConoziZ is a solid generated by the revolving of a parabola or 
hyperbola around its axis. 

A ,S;?/iero2(^ is a solid generated in like manner to a conoid by an 
ellipse. 

To construct a Parabola— fig. 1. 
B 




CONIC SECTIONS, 



55 



Draw an isosceles triangle, A B D, whose base shall be equal to 
that of the proposed parabola, and its altitude, C B, twice that of it. 

Divide each side, A B, D B, into 8, or any number of equal parts ; 
then draw lines 1 1, 2 2, 3 3, &c., and their intersection will define 
the curve of a parabola. 

Note. — The following figures are drawn to a scale of 100 parts to an inch. 

To construct an Hyperbola,* the Transverse and Conjugate Diameters 
being given— fig. 2. 




Make A B the transverse diameter, and C D, perpendicular to it, 
the conjugate. 

Bisect A B in 0, and from 0, with the radius O C or D, de- 
scribe the circle D / c F, cutting A B produced in F and /, which 
points will be the foci. 

In A B produced take any number of points, n n, &c., and from 
F and /, as centres, with B 72, A tz, as radii, describe arcs, cutting 
each other in s s, &c. 

Through s 5, &c., draw the curve s B s, and it will be the hyper- 
bola required. 




^ To describe hyperbolas by another metliod, see Gregory's Mathematics, p. 160. 



56 CONIC SECTIONS. 

To find the length of the Ordinate, E F, of an Ellipse, the Transverse, A 
B, Conjugate, C D, and Abscissa, A F and F B, heing known— fig. 3. 

Rule. — As the transverse diameter is to the conjugate, so is the 
square root of the product of the abscissae to the ordinate which 
divides them. 

Example.— The transverse axis, A B, is 100 ; the conjugate, C D, 
is 60; one abscissa, B F, is 20 ; the other, A F, is (100— 20) = 80. 

100 : 60 : : v'^OxSO : 24 Ans. 

The Transverse and Conjugate diameters, and an Ordinate heing known, 
to find the Ahscissce — fig. 3. 

Rule. — As the conjugate diameter is to the transverse, so is the 
square root of the difference of the squares of the ordinate and semi- 
conjugate to the distance between the ordinate and centre ; and 
this distance, being added to and subtracted from the semi-trans- 
verse, will give the abscissas required. 

Example.— The transverse diameter, A B, is 100 ; the conjugate, 
C D, is 60 ; and the ordinate, F E, is 24. 

60 : 100 : : ^ 24^—302 . 39^ distance between the 

ordinate and centre ; 
then lOOH-2— 30 =:20, one abscissa ; 

100-^2+30 = 80, the other abscissa. 

When the Conjugate, Ordinate, and Ahscissce are known, to find the 
Transverse— fig. 3. 

Rule. — To or from the semi-conjugate, according as the greater 
or less abscissa is used, add, or subtract the square root of the dif- 
ference of the squares of the ordinate and semi-conjugate. Then, 
as this sum or difference is to the abscissa, so is the conjugate to 
the transverse. 

Example. — The ordinate, F E, is 24 ; the less abscissa, F B, is 
20 ; and the conjugate, C D, is 60. 



30—^242-302 — 12 ; 
then 12 : 20 : : 60 : 100 Ans, 

The Transverse^ Ordinate, and Ahscissce heing given, to find the Con- 
jugate—fig. 3. 

Rule. — As the square root of the product of the abscissae is to the 
ordinate, so is the transverse diameter to the conjugate. 

Example. — The transverse is 100, the ordinate 24, one abscissa 
20, the other 80. 

-v/80X20 : 24 : : 100 : 60 Ans. 

PARABOLAS. 

Any three of the four following terms heing given, viz., any two Or- 
dinates and their Abscissce, to find the fourth — fig. 4. 
Rule. — As any abscissa is to the square of its ordinate, so i^ ■^R" 
other abscissa to the square of its ordinate. 



CONIC SECTIONS. 

4. e 



57 




Example. — The abscissa, e g, is 50, its ordinate, c g, 35.35 ; re- 
quired the ordinate A F, whose abscissa, e F, is 100. 
50 : 35.35=^ : : 100 : ^2500 r=r 50 Ans. 

HYPERBOLAS. 
B 




When the Transverse, the Con jit gale, and the less Abscissa, B n, are 
given, to find an Ordinate, e n — fig. 5. 
Note. — In hyperbolas, the less abscissa, added to the axis, gives the greater. 

Rule. — As the transverse diameter is to the conjugate, so is the 
square root of the product of the abscissae to the ordinate required. 

When the Transverse, the Conjugate, and an Ordinate are given, to 
find the AhscisscB — fiig. 5. 

Rule. — To the square of half the conjugate add the square of the 
ordinate, and extract the square root of that sum. 

Then, as the conjugate diameter is to the transverse, so is the 
square root to half the sum of the abscissae. 

To this half sum add half the transverse diameter for the greater 
abscissa, and subtract it for the less. 

When the Transverse^ the Abscissce, and Ordinate are given, to find 
the Conjugate — fig. 5. 
Rule. — As the square root of the product of the abscissae is to 
the ordinate, so is the transverse diameter to the conjugate. 

When the Conjugate, the Ordinate, and the Abscissce are given, to find 
the Transverse — fiig. 5. 

Rule. — Add the square of the ordinate to the square of half the 
conjugate, and extract the square root of that sum. 

To this root add half the conjugate when the less abcissa is used, 
and subtract it when the greater is used, reserving the difference 
or sum. 

Then, as the square of the ordinate is to the product of the ab- 
scissa and conjugate, so is the sum, or difference above found, to 
the transverse diameter. 



58 CONIC SECTIONS. 

Examples. — In the hyperbola, /^5. 2 and 5, the transverse dian 
eter is 100, the conjugate 60, and the abscissa, B n, is 40 ; required 
the ordinate e n. 

100 : 60 : : v^(40+100x40) = 74.8 : 44.8 Ans. 

The transverse is 100, the conjugate 60, and ordinate e n, 44.8 
what are the abscissas'? Ans. 40 and 140. 

The transverse is 100, the ordinate 44.8, the abscissas 140 and 40 
what is the conjugate 1 Ans. 60. 

The conjugate is 60, the ordinate 44.8, and the less abscissa 40 
what is the transverse 1 A7is. 100. 



MENSURATION OF SURFACES. 



59 



MENSURATION OF SURFACES. 




OF FOUR-SIDED FIGURES 
2. a '■ 




To find the Area of a four-sided Figure, v)hether it be a Square, Paral- 
lelogram, Rhombus, or a Rhomboid. 
Rule. — Multiply the length by the breadth or perpendicular height, 
and the product will be the area. 

OF TRIANGLES. 

To find the Area of a Triangle^figs. 5 and 6, 

r c 

6. 




Rule. — Multiply the base a b hy the perpendicular height c d^ 
and half the product will be the area. 

To find the Area of a Triangle by the length of its sides. 
Rule. — From half the sum of the three sides subtract each side 
separately ; then multiply the half sum and the three remainders 
continually together, and the square root of the product will be the 
area. 

To find the Length of one side of a Right-angled Triangle, wtien the 
Length of the other two sides are given — fig. 7. 

Rule. — To find the hypothenuse a c. Add together the square 
of the two legs a b and a c, and extract the square root of that sum. 

To find one of the legs. Subtract the square of the leg, of which 
the length is known, from the square of the hypothenuse, and the 
square root of the difference will be the answer. 

Note. — For Spherical Triangles, see page 68. 

OF TRAPEZIUMS AND TRAPEZOIDS. 

To' find the Area of a Trapezium — fig. 8. 
Rule. — Multiply the diagonal a c by the sum of the two perpen- 
diculars falling upon it from the opposite angles, and half the product 
will be the area. 



60 



MENSURATION OF SURFACES. 





To find the Area of a Trapezoid— fig. 9. 

Rule. — Multiply the sura of the parallel sides a h, d c, by a h, the 
perpendicular distance between them, and half the product will be 
the area. 

OF REGULAR POLYGONS. 

Rule.— Multiply half the perimeter of the figure by the perpen- 
dicular, falling from its centre upon one of the sides, and the prod- 
uct will be the area. 

To find the Area of a Regular Polygon, when the side only is given. 

Rule. — ^^lultiply the square of the side by the multiplier opposite 
to the name of the polygon in the following table, and the product 
will be the area. 



No. of 

Sides. 


Name of Polygon. 


Angle. 


Angle of 
Folygnn. 


Area. 


A 


B 


c 


3 


Trigon 


120° 


60^ 


0.433012 


2. 


1.732 


.5773 


4 


Tetragon 


90 


90 


1.000000 


1.41 


1.414 


.7071 


5 


Pentagon 


72 


108 


1.720477 


1.238 


1.175 


.8506 


6 


Hexagon > 


60 


120 


2.598076 


1.156 


—Radius 


C =:l'gth 

( of side 


7 


Heptagon 


51^ 


128i 


3.633912 


1.11 


.8677 


1.152 


8 


Octagon 


45 


135 


4.828427 


1.08 


.7653 


1.3065 


9 


No n agon 


40 


140 


6.181824 


1.06 


.6840 


1.4619 


10 


Decagon 


36 


144 


7.694208 


1.05 


.6180 


1.6180 


11 


Qndecagon 


32fy 


147TT 


9.365640 


1.04 


.5634 


1.7747 


12 


Dodecagon 


1 30 


150 


11.196152 


1.037 


.5176 


1.9318 



Additional uses of the foregoing Table. 

The third and fourth columns nf the table will greatly facilitate the construction 
of these figures, with the aid of the sector. Thus, if it is required to describe an oc- 
tagon, opposite to it, in column third, is 45 ; then, with the chord of 60 on the sector 
as radius, describe a circle, taking the length 45 on the same line of the sector ; 
mark this distance off on the circumference, which, being repeated around the circle, 
will give the points of the sides. 

The fourth column gives the angle which any two adjoining sides of the respective 
figures make with each other. 

Take the length of a perpendicular drawn from the centre to one of the sides of a 
polygon, and multiply this by the numbers in column A, the product will be the ra- 
dius of the circle that contains the figure. 

The radius of a circle multiplied by the number in column B, will give the length 
of the side of the corresponding figure which that circle will contain. 

The length of the side of a pohgon multiplied by the corresponding number in the 
column C, will give the radius of the circumscribing circle. 



MENSURATION OF SURFACES 



61 



OF REGULAR BODIES. 



To find the Superficies of any Regular Body. 
Rule. — Multiply the tabular surface in the following table by the 
square of the linear edge, and the product will be the superficies. 



Number of Sides. 


Names. 


Surfaces. 


4 


Tetrahedron 


1.73205 


6 


Hexahedron. 


6.00000 


8 


Octahedron. 


3.46410 


12 


Dodecahedron. 


20.64573 


20 


Icosahedron. 


8.66025 



OF IRREGULAR FIGURES. 




Tofmd the Area of an Irregular Polygon, abcdefg—fig. 10. 
Rule. — Draw diagonals to divide the ligure into trapeziums and 
triangles ; find the area of each separately, and the sum of the whole 
will give the area required. 

To find the Area of a Long Irregular Figure, bdca—fig. 11. 
Rule. — Take the breadth in several places, and at equal distan- 
ces apart ; add them together, and divide the sum by the number 
of breadths for the mean breadth ; then multiply that by the length 
of the figure, and the product will be the area. 



OF CIRCLES. 




14. 



6 

A/' 


rr^ 


' 1 ^^ 


■^^ 




/ 


o 


\ 


\^ 




q\ 










'^\ 








hi. 



To find the Diameter and Circumference of any Circle. 
Rule 1.— Multiply the diameter by 3.1416, and the product will 
be the circumference. 

F 



62 MENSURATION OF SURFACES. 

Rule 2.— Divide the circumference by 3.1416, and the quotient 
will be the diameter. 

Rule 3.— Or, as 7 is to 22, so is the diameter to the circumfer- 
ence. 

Or, as 22 is to 7, so is the circumference to the diameter. 

Or, as 1 13 is to 355, so is the diameter to the circumference, &c 

To find the Area of a Circle. 

Rule 1.— Multiply the square of the diameter by .7854, or the 
square of the circumference by .07958, and the product will be the 
area. 

Rule 2.— Multiply half the circumference by half the diameter. 

Rule 3.— As 14 is to 11, %o is the square of the diameter to the 
area , or, as 88 is to 7, so is the square of the circumference to the 
area. 

To find the Length of any Arc of a Circle— fig. 12. 

Rule 1.— From 8 times the chord of half the arc a.c, subtract the 
chord ab of the whole arc ; one third of the remainder will be the 
length nearly. 

Rule 2.— Multiply the radius ao of the circle by .0174533, and 
that product by the degrees in the arc. 

Rule 3.— As 180 is to the number of degrees in the arc, so is 
3.1416 times the radius to its length. 

1. When the Chord of the Arc and the Versed Sine of half the Arc are 

given. 

Rule 4.— To 15 times the square of the chord ab, add 33 times 
the square of the versed sine c d, and reserve the number. 

To the square of the chord add 4 times the square of the versed 
sine, and the square root of the sum will be twice the chord of half the 
arc. 

Multiply twice the chord of half the arc by 10 times the square of 
the versed sine, divide the product by the reserved number, and add 
the quotient to twice the chord of half the arc : the sum will be the 
length of the arc very nearly. 

Note.— 1. diameter X .8862 = side of an equal square. 

2. circumference X .2821= " " " 

3. diameter X .7071 = " of the inscribed square. 

4. circumference X .2251 = " " " 

5. area X .9003= " " " 

6. side of a square XI. 4142 = diam. of its circums. circle. 

7. " " X4.443 =circum. " " 

8. " " Xl.128 = diam. of an equal circle. 

9. <' " X 3.545 =circum. " " 
10. square inches X 1-273 = round inches. 

When the Chord of the Arc, and the Chord of half the Arc are given. 
Rule 5.— From the square of the chord of half the arc subtract 

Note.— If the length for any number of degrees, minutes, &c., is required (see 
page 67 for the units, radius being 1), multiply them by the number of degrees, &c. 
in the arc, and the answer is the length. 



MENSURATION OF SUEFACES. 63 

the square of half the chord of the arc, and the remainder will be 
the square of the versed sine : then proceed as above. 

Note. — The chord of half the arc is equal to the square root of the sum of the 
squares of the versed sine or height, and half the chord of the entire arc. 

When the Diameter and the Versed Sine of half the Arc are given. 

Rule 6. — From 60 times the diameter co, subtract 27 times cd 
the versed sine, and reserve the number. 

Multiply the diameter by the versed sine, and the square root of 
the product will be the chord of half the arc. 

Multiply twice the chord of half the arc by 10 times the versed 
sine, divide the product by the reserved number, and add the quo- 
tient to twice the chord of half the arc : the sum will be the length 
of the arc very nearly. • 

Note. — When the diameter and chord of the arc are given, the versed sine may 
be found thus : From the square of the diameter subtract the square of the chord, 
and extract the square root of the remainder. Subtract this root from the diameter, 
and half the remainder will give the versed sine of half the arc. 

The square of the chord of half the arc being divided by the diameter, vi^ill give the 
versed sine ; or, being divided by the versed sine, vv^iU give th.e diameter. 

To find the Area of a Sector of a Circle — fig. 13. 

Rule 1. — Multiply the length of the arc adb by half the length 
of the radius ao. 

Rule 2. — As 360 is to the degrees in the arc of the sector, so is 
the area of the circle to the area of the sector. 

Note. — If tlie diameter or radius is not given, add the square of half the chord of 
the arc to the square of the versed sine of half the arc ; this sum being divided by the 
versed sine, will give the diameter. 

To find the Area of a Segment of a Circle— fig. 12. 
(See table of Areas, page 72.) 

Rule 1. — Find the area of the sector having the same arc with 
the segment, then find the area of the triangle formed by the chord 
of the segment and the radii of the sector, and the difference of 
these areas, according as the segment is greater or less than a 
semicircle, will be the area required. 

Rule 2.— To the chord ah oi the whole arc, add the chord ac of 
half the arc, and I of it more ; then multiply the sum by the versed 
sine c d, and y\ of the product will be the area. 

Rule 3.— Multiply the chord of the segment by the versed sine, 
divide the product by 3, and multiply the remainder by 2. 

Cube the height, find how often twice the length of the chord is 
contained in it, and add the quotient to the former product, and it 
will give the area nearly. 

To find the Area of a Circular Zone— fig. 14. 
(See table of Areas, page 80.) 
Rule 1. — When the zone is less than a semicircle. To the area of the 
trapezoid ahcd add the area of the segments ab, cd; their sum is 
the area. 



64 



MENSURATION OF SURFACES. 



Rule 2. — When the zone is greater than a semicircle. To the area of 
the parallelogram bgdh, add the area of the segments big, dkh; 
their sum is the area. 

To find the Convex Surface of any Zone or Segment— figs. 38 and 39. 
Rule.— Multiply the height c b, or b d, of the zone or segment by 
the circumference of the sphere, and the product is the surface. 

OF UNGULAS. 
To find the Convex Surface of the Ungulas—figs. 27, 28, 29, and 30. 

Rules.— For fig, 27, multiply the length of the arc line abc of the 
base by the height ad. ^ 

For fig. 28, multiply the circumference of the base of the cylinder 
efg by half the sum of the greater and less lengths a e, cf 

For fig. 29, multiply the sine ad, of half the arc ag, of the base 
a eg, by the diameter eg of the cylmder, and from this product sub- 
tract the product* of the arc age and cosine df Multiply the dif- 
ference thus found by the quotient of the height g b, divided by the 
versed sine e d. 

For fig. 30 (conceive the section to be continued till it meets the 
side of the cylinder produced), then find the surface of each of the 
ungulas thus formed, and their difference is the surface required. 

Note. — For rules to ascertain the surface of conical ungulas, see Ryan's Bonny- 
costless Mensuration, page 136 (1639). 

To find the Area of a Circular Ring or Space included between two 
Concentric Circles— fig. 54. 

Rule.— Find the areas of the two circles ad, be separately, and 
their difference will be the area of the ring. 




OF ELLIPSES. 
16. 




To find the Circumference of an Ellipse— fig. 15. 

I^uLE. — Square the two axes ah and cd, and multiply the square 
root of half their sum by 3.1416 ; the product will be the circumfer- 
ence. 

To find the Area of an Ellipse— fig. 15. 

Rule. — Multiply the two diameters together, and the product by 

.7854. 



* When this product exceeds the other, add them together, and when the cosine is 
0, the product is 0. 



MENSURATION OF SURFACES. 



65 



To find the Area of an Elliptic Segment, <^^g—fig- 16. 
Rule. — Divide the height of the segment a;? by the axis ah, of 
which it is a part, and find in the table of circular segments, page 
72, a segment having the same versed sine as this quotient ; then 
multiply the segment thus found and the two axes of the ellipse to- 
gether, and the product will give the area. 



17. 



Uh 



OF PARABOLAS 

18. 



r 




To find the Area of a Parabola — fig. 17. 
Rule. — Multiply the base dfhy the height ^e, and | of the prod- 
uct will be the area. 

To find the Area of a Frustrum of a Parabola — fig. 17. 
Rule. — Multiply the difference of the cubes of the two ends of 
the frustrum acdf by twice its altitude b e, and divide the product 
by three times the difference of the squares of the ends. 

To find the Length of a Parabolic Curve cut off by a Double Ordinate — 

fig. 18. 
Rule. — To the square of the ordinate a b add A of the square of 
the abscissa c b ; the square root of that sum, multiplied by 2, will 
give the length of the curve nearly. 



OF HYPERBOLAS 
20. 





To find the Area of a Hyperbola — fig. 19. 
Rule. — To the product of the transverse and abscissa add | of the 
square of the abscissa a b, and multiply the square root of the sum 
by 21. Add 4 times the square root of tHe product of the transverse 
and abscissa to the product last found, and divide the sum by 75. 

Divide 4 times the product of the conjugate and abscissa by the 
transverse, and this last quotient, multiplied by the former, will give 
the area nearly. 

F2 



QQ MENSURATION OF SURFACES. 

To find the Length of a Hyperbolic Curve— fig. 20. 
Rule.— As the transverse is to the conjugate, so is the conjugate 
to the parameter. To 21 times the parameter of the axis add 19 
times the transverse, and to 21 times the parameter add 9 times the 
transverse, and multiply each of these sums by the quotient of the 
abscissa b a, divided by the transverse. To each of these tv^^o prod- 
ucts add 15 times the parameter, and divide the former by the lat- 
ter ; multiply this quotient by the ordinate, and the product is the 
length of half the curve nearly. 

OF CYLINDRICAL RINGS. 
To find the Convex Surface of a Cylindrical Ring— fig. 54. 
I^uiE.— To the thickness of the ring ab add the inner diameter 
c ; multiply this sum by the thickness, and the product by 9.8696, 
nd it will give the surface required. 

To find the Area of a Circular Ring— fig. 54. 
Rule —The difference of the areas of the two circles will be the 
area of the ring. 

OF LUNES. 



To find the Area of a Lune—fig. 21. 
Rule.— Find the areas of the two segments a deb, abce from 
which the lune is formed, and their difference will be the area re- 
quired.* 



OF CYCLOIDS. 
s 




n 

To find the Area of a Cycloid— fig. 22. 
Rule.— Multiply area of generating circle a 6 c by 3, and the prod- 
uct is the area. 

* If semicircles be described on the three sides of a right-angled triangle as diame- 
ters, two iunes will be formed, their united areas being equal to the area ol tne wi 
angle. 



MENSURATION OF SURFACES. 6*7 

OF CYLINDERS. 

To find the Convex Surface of a Cylinder— fig. 25. 

Rule. — Multiply the circumference by the length, and the prod- 
uct will be the surface. 

OF CONES OR PYRAMIDS. 
To find the Convex Surface of a Right Cone or Pyramid — figs. 31 and 33. 
Rule. — Multiply the perimeter or circumference of the base by 
the slant height, and half the product will be the surface. 

To find the Convex Surface of a Frustrum of a Right Cone or Pyramid — 
figs. 32 and 34. 

Rule. — Multiply the sum of the perimeters of the two ends by the 
slant height or side, and half the product will be the surface. 

OF SPHERES. 

To find the Convex Surface of a Sphere or Globe— fig. 37. 

Rule — Multiply the diameter of the sphere by its circumference, 
and the product is the surface. 

OF CIRCULAR SPINDLES. 
To find the Convex Surface of a Circular Spindle — fig. 45. 
Rule. — Multiply the length /c of the spindle by the radius oc of 
the revolving arc; multiply the said arc fac by the central distance 
oe, or distance between the centre of the spindle and centre of the 
revolving arc. Subtract this last product from the former, double 
the remainder, multiply it by 3.1416, and the product is the surface. 

Note. — The same rule will serve for any zone or seg-ment, cut off perpendicularly 
to the chord of the re-volving arc ; in this case, then, the particular length of the 
part, and the part- of the arc which describes it, must be taken, in lieu of the whole 
length and whole arc. 



BY MATHEMATICAL FORMUL.'E. 

LINES. CIRCLE. 

Ratio of circumference to diameter, ^ = 3.1416. 

T -L r ^^' ^ 

Length of an arc = — - — nearly ; c the chord of the arc, and c' 

o 

the chord of half the arc. 

Length of 1 degree, radius being 1, = .0174533 
" 1 minute, = .0002909 

" 1 second, = .0000048 

ELLIPSE. 
Circum/drcTicfi =l||p^J(a2+^>2) nearly, a and h being the axes.' 

PARABOLA. 

Length of an arc^ commencing at the vertex, =s \/(-q"+J') near- 
ly, a being the abscissa, and h the ordinate. 



68 



MENSURATION OF SURFACES. 



QUADRILATERALS. 

Half the product of the diagonals X the sine of their angle. 

CIRCLE, 

I>r2 ; or diam. 2 x. 78539816 ; or circum. ^ X. 0795774. 

CYLINDER. 

Curved-surface =: height X perimeter of base. 

SPHERICAL ZONE OR SEGMENT. 

2prh ; or, the height of the zone or segment X the circumference 
of the sphere. 

CIRCULAR SPINDLE. 

2p{rc—a^/r-—ic^) ; a being the length of the arc, and c its chord, 
or the length of the spindle. 

SPHERICAL TRIANGLE. 

pr^ ^'" ^ ; s being the sum of the three angles. 
180 

ANY SURFACE OF REVOLUTION. 

2prXl; or, the length of the generating element x the circum 
ference described by its centre of gravity. 

Illustrations.— Let abcbe the side of a cylinder, br the radius ; 
then abcis the generating element, b the centre of gravity (of the 
line), and b r the radius of the circle described by abc. 



Then, ifa6c = 10, 5r=:5; 10x(5+5X 
3.1416) = 314.16. 




Parabola. 



acX{^brXp), p being in this and all other 
instances = 3.1416, b the centre of gravity, 
and b r the radius of its circumference. 



MENSURATION OF SURFACES. 



69 



Or, take a umform piece of board or thick pasteboard, and cut out 

^nH £'M fi^'* ^'^ ^'■'^'' '"'^•J"''"^'' ' ^^'eh both pieces together 
and then the figure separately, and say, as the gross weight is to the 
entire surface, so is the weight of the figure to its surface. 

CAPILLARY TUBE. 

Let the tube be weighed when empty, and again when filled with 
mercury ; let «, be the difference of thos^ weights in troy grains Tnd 
/the length of the tube in inches. " "oy grains, ana 



Diameter = .019353^/—. 



In which 
Thenp 



USEFUL FACTORS, 
p represents the Circwitiference of a Circle whose Diameter is 1. 



4? 

iP 
iP 

iP 
iP 

ip 

3F0-P 

1 



= 3.1415926535897932384626+ j 

= 6.283185307179+ 

= 12.566370614359+ 

= 1.570796326794+ 

= 0.785398163397+ 

= 4.188790 

= .523598 

= .392699 

= .261799 

= .008726 

= .318309 



= .636619 



1.273239 
.079577 



4p 

Vp = 
Wp = 

4' 

360 

- - =114.591559 

|i» = 2.094395 

- = 1.909859 
P 
36p =113.097335 



1.772453 

.886226 
3.544907 

.797884 
.564189 



70 MENSURATION OF SURFACES. 



Examples in Illustration of the foregoing Rules. 

Required the area, 1. Of the rhombus, fig. 3, a c 12 feet 6 inches, and its he^ 
aZ>, 9 feet 3 inches. Ans. 115.625 feet. 

2. Of the ti'iangle a J c, fig. 5, a 5 being 10 feet, and cd o feet. Ans. 25 feet. 

3. Of the triangle abc, fig. 7, its three sides measuring respectively 24, 36, and 
48 feet. ^715. 418.282. 

4. In the right-angled triangle a 6 c, fig. 7, the base is 56, and the height 33 ; what 
is the hypolhenuse 1 Ans. 65. 

5. If the hypothenuse of a triangle be 53, and the base 45, what is the perpendic- 
ular ? Ans. 28. 

6. Required the area of the trapezium, fig. 8, the diagonal ac 84, the perpendicu- 
lars 21 and 28. Ans. 2058. 

7. Of the trapezoid, fig. 9, a & 10, <Z c 12, and ah& feet. ^715. 66. 

8. Of an octagon, the side being 5. 52 = 25X4.828427 = 120.710675 Ans. 

9. The length of a perpendicular from the centre to one of the sides of an octa- 
gon is 12 ; what is the radius of the circumscribing circle 1 

12X1.08 (table, page 60) = 12.96 Ans. 

10. The radius of a circle being 12.96, what will be the length of one side of an 
inscribed octagon 1 12.96X .765 (page 60) — 9.914 Ans. 

11. The length of the side of a decagon is 10 ; what is the radius of the circum- 
scribing circle 1 10X1.618 (page 60) = 16.18 Ans. 

12. The chord a &, fig. 12, is 48, and the versed sine cd I'd', what is the length 
of the arc 1 

By Rule 4, twice the chord of half the arc is 60 , then 60-^2 = 30, chord of half 
the arc, and 30x8 = 240—48 — 192-^3 = C4 Ans. 

13. The diameter c 0, fig. 12, is 50, and the versed sine cc^ 18 ; what is the length 
of the arc ? 

By rule 6 . . . 50 X 60-f 18X27 = 2514v/50X 18 = 900 = 30. 
Then 30X2 = 60X10X18 = 10800-f-2514 = 4.2959+30X2 = 64.2959 ^715. 

14. The diameter of a circle is 50, and the chord of half the arc 30 ; what is 
the length of the arc % Ans. 64.2959. 

15. What is the aiea of a sector, the chord of the arc being 40, and the versed 
sine 15 1 -^715. 558.125. 

16. The radius of a sector b, fig. 13, is 20, and the degrees in its arc 22 ; what is 
the area 1 Ans. 76.7947. 

17. The radius oc is 10, and the chord ac 10; w^hat is the area of the segment 
acbd,ng.l21 Ans. 52.36. 

18. The greater chord, b d, fig. 14, is 96, the lesser, a c, 60, and the breadth 26 ; 
what is the area of the zone ? Ans. 2136.75. 

19. The sine of half the arc, fig. 29, is 7, the diameter of the cylinder 15, the co- 
sine on eg, at the intersection of a c, 2.7, the versed sine 4.8, and the height, i^, 12 ; 
what is the convex surface 1 Ans. 196. 

20. The height, ap, of an elliptic segment, fig. 16, is 10, and the axes 25 and 35 
respectively ; what is the area 1 

10-i-35= .2857 tabular versed sine, and segment = .185153X35X25 = 162.0088 

^715. 

21 . In the parabolic frustrum, a c df, fig. 17, the ends a c and df are 6 and 10, and 
the height be is 5; what is the area 1 

■\f)i R3 704 

iP=65 = 64 = 12-^><« °^ ^ = '"-^ "'"■ 

22. The abscissa c b, fig. 18, is 12, and its ordinate a 6 6 ; what is the length of 
acd? ^715.30.198. 

23. The transverse and conjugate diameters of a hyperbola, fig. 19, are 100 and 
60, and the abscissa a 6 60 ; what is the area ? Ans. 4320. 

24. What is the curve a c d of the hyperbola, fig. 20, the abscissa a 6 40 "? 

Ans. 59.85. 



MENSURATION OF SURFACES. 71 

25. The chord ac, fig. 21, is 19, the heights ed 6.9, and c6 2.4 ; what is the area 
ot tne lune . ^^^ g^ 3 

tbf; J]^^/A^''5'f '"" ''''''^^ ''*'' ^S- ^' ^^ ^ ^"^^^' diameter ; what is the area of 
me cycloid bcdl ^^^ 37.6992. 

conve Js^urfaTeS""^ ^ '''''^' ^°' ^^' '' ^ ^'^^' ^"""^ *^^ '^^''* ^^'^^^ ^^ ^^^^ ' ^^^* ^^ t^« 

.^W5u 70.686. 

15?; Ji"^ thickness of a cylifidric ring, fig. 54, is 3 inches, and the inner diameter 
12 mches ; what is the convex surface ? ^„5. 444.132. 

29. What is the convex surface of a globe, fig. 37, 17 inches in diameter 1 

^715. 6.305 square feet. 
rnH?.;.^^'''?n^ ^^i""""^^^® ^f i^^ circular spindle, fig. 45, the length fc 14.142, the 
radius o c 10, and the central distance o e 7.071 inches ? Anl 190.82 inches 

31. What is the surface of an octahedron, the linear side being 2 inches ? 

22x3.46410 (tabular surface) = 13.85640 \/fnff. 



72 



AREAS OF THE SEGMENTS OF A CIRCLE. 



Table of the Areas of the Segments of a Circle., the diameter of which 
is Unity, and supposed to he divided into 1000 equal Parts. 



^sIlS" ^^=-'^-- 'is' ^^^•^^-• 



.00004 

.00011 

.00021 

.00033 

.00047 

.00061 

.00077 

.00095 

.00113 

.00132 

.00153 

.00174 

.00196 

.00219 

.00243 

.00268 

.00294 

.00320 

.00347 

.00374 

.00403 

.00432 

.00461 

.00492 

.00523 

.00554 

.00586 

.00619 

.00652 

.00686 

.00720 

.00755 

.00791 

.00827 

.00863 

.00900 

.00938 

.00976 

.01014 

.01053 

.01093 

.01133 

.01173 

.01214 

.01255 

.01297 

.01339 

.01381 

.01424 

.01468 

.01511 

.01556 

.01600 

.01645 



.055 

.056 

.057 

.058 

.059 

.060 

.061 

.062 

.083 

.064 

.065 

.066 

.067 

.068 

.069 

.070 

.071 

.072 

.073 

.074 

.075 

.076 

.077 

.078 

.079 

.080 

.081 

.082 

.083 

.084 

.085 

.086 

.087 

.088 

.089 

.090 

.091 

.092 

.093 

.094 

.095 

.096 

.097 

.098 

.099 

.100 

.101 

.102 

.103 

.104 

.105 

.106 

.107 

.108 



.01691 

.01736 

.01783 

.01829 

.01876 

.01923 

.01971 

.02019 

.02068 

.02116 

.02165 

.02215 

.02265 

.02315 

.02365 

.02416 

.02468 

.02519 

.02571 

.02623 

.02676 

.02728 

.02782 

.02835 

.02889 

.02943 

.02997 

.03052 

.03107 

.03162 

.03218 

.03274 

.03330 

.03.387 

.03444 

.03501 

.03558 

.03616 

.03674 

.03732 

.03790 

.03849 

.03908 

.03968 

.04027 

.04087 

.04147 

.04208 

.04268 

.04329 

.04390 

.04452 

.04513 

.04575 



Versed 
Sine. 


Scg. Area. 


Versed 
Sine. 


.109 


.04638 


.163 


.110 


•.04700 


.164 


.111 


.04763 


.165 


.112 


.04826 


.166 


.113 


.04889 


.167 


.114 


.04952 


.168 


.115 


.05016 


.169 


.116 


.05080 


.170 


.117 


.05144 


.171 


.118 


.05209 


.172 


.119 


.05273 


.173 


.120 


.05338 


.174 


.121 


.05403 


.175 


.122 


.05468 


.176 


.123 


.05534 


.177 


.124 


.05600 


.178 


.125 


.05666 


.179 


.126 


.05732 


.180 


.127 


.05799 


.181 


.128 


.05865 


.182 


.129 


.05932 


.183 


.130 


.05999 


.184 


.131 


.06067 


.185 


.132 


.06134 


.186 


.133 


.06202 


.187 


.134 


.06270 


.188 


.135 


.06338 


.189 


.136 


.06407 


.190 


.137 


.06476 


.191 


.138 


.06544 


.192 


.139 


.06614 


.193 


.140 


.06683 


.194 


.141 


.06752 


.195 


.142 


.06822 


.196 


.143 


.06892 


.197 


.144 


.06962 


.198 


.145 


.07032 


.199 


.146 


.07103 


.200 


.147 


.07174 


.201 


.148 


.07245 


.202 


.149 


.07316 


.203 


.150 


.07387 


.204 


.151 


.07458 


.205 


.152 


.075.30 


.206 


.153 


.07602 


.207 


.154 


.07674 


.208 


.155 


.07746 


.209 


.156 


.07819 


.210 


.157 


.07892 


.211 


.158 


.07964 


.212 


.159 


.08038 


.213 


.160 


.08111 


.214 


.161 


.08184 


.215 


.162 


.08258 


.216 



AREAS OF THE SEGMENTS OF A CIRCLE. 



Table — (Continued). 



Versed 
Sine, 


Seg. Area. 


Versed 
Sine. 


Ses- Area. 


^'ersed 
Siue. 


Seg. Area. 


Versed 
Sine. 


Seg. 


.217 


.12563 


.272 


.17286 


.327 


.22321 


.382 


,2" 


.218 


.12645 


.273 


.17375 


.328 


.22415 


.383 


.2"; 


.219 


.12728 


.274 


.17464 


.329 


.22509 


.384 


.2; 


.220 


12811 


.275 


.17554 


.330 


.22603 


.385 


.27 


.221 


.12894 


.276 


.17643 


.331 


.22697 


.386 


.27 


.222 


.12977 


.277 


.17733 


.332 


.22791 


.387 


.2S 


.223 


.13060 


.278 


.17822 


.333 


.22885 


.388 


.2S 


.224 


.13143 


.279 


.17912 


..334 


.22980 


.389 


.2S 


.225 


.13227 


.280 


.18001 


.335 


.23074 


.390 


.28 


226 


.13310 


.281 


.18091 


.336 


.23168 


.391 


.28 


227 


.13394 


.282 


.18181 


.337 


.23263 


.392 


.28 


228 


.13478 


.283 


.18271 


.338 


.23358 


.393 


.28 


229 


.13562 


.284 


.18361 


.339 


.23452 


.394 


.28 


230 


.13646 


.285 


. 18452 


.340 


.23547 


.395 


.28 


231 


.13730 


.286 


. 18542 


.341 


.23642 


.396 


.28 


232 


.13815 


.287 


.18632 


.342 


.23736 


.397 


.29 


233 


.13899 


.288 


.18723 


.343 


.23831 


.398 


.29 


234 


.13984 


.289 


.18814 


.344 


.23926 


.399 


.29 


235 


.14068 


.290 


.18904 


.345 


.24021 


.400 


.29 


236 


.14153 


.291 


.18995 


.346 


.24116 


.401 


.29 


237 


.14238 


.292 


.19086 


.347 


.24212 


.402 


.29 


238 


.14323 


.293 


.19177 


.348 


.24307 


.403 


.29 


239 


.14409 


.294 


.19268 


.349 


.24402 


.404 


.29 


240 


. 14494 


.295 


.19359 


.350 


.24498 


.405 


.29 


241 


.14579 


.296 


.19450 


.351 


.24593 


.406 


.29 


242 


.14665 


.297 


.19542 


.352 


.24688 


.407 


.30 


243 


.14751 


.298 


.19633 


.353 


.24784 


.408 


.30 


244 


.14837 


.299 


.19725 


.354 


.24880 


.409 


.30 


245 


. 14923 


.300 


.19816 


.355 


.24975 


.410 


.30, 


246 


.15009 


.301 


.19908 


.356 


.25071 


.411 


.30^ 


247 


.15095 


.302 


.20000 


.357 


.25167 


.412 


.30. 


248 


.15181 


.303 


.20092 


.358 


.25263 


.413 


.30( 


249 


.15268 


.304 


.20184 


.359 


.25359 


.414 


.30' 


250 


.15354 


.305 


.20276 


.360 


.25455 


.415 


.30^ 


251 


.15441 


.306 


.20368 


.361 


.2555] 


.416 


.30j 


252 


.15528 


.307 


.20460 


.362 


.25647 


.417 


.311 


253 


.'5614 


.308 


.20552 


.363 


.25743 


.418 


.31 


254 


.15701 


.309 


.20645 


.364 


.25839 


.419 


.311 


255 


.15789 


.310 


.20737 


.365 


.25935 


.420 


.3n 


256 


.15876 


.311 


.20830 


.366 


.26032 


.421 


.31^ 


257 


.15963 


.312 


.20922 


.367 


.26128 


.422 


.31." 


258 


.16051 


.313 


.21015 


.368 


.26224 


.423 


.3U 


259 


.16138 


.314 


.21108 


.369 


.26321 


.424 


.31( 


260 


.16226 


.315 


.21201 


.370 


.26417 


.425 


.317 


261 


.16314 


.316 


.21294 


.371 


.26514 


.426 


.316 


262 


.16401 


.317 


.21387 


.372 


.26611 


.427 


.3U 


263 


.16489 


.318 


.21480 


.373 


.26707 


.428 


.32C 


264 


.16578 


.319 


.21573 


.374 


.26804 


.429 


.32: 


265 


.16666 


.320 


.21666 


.375 


.26901 


.430 


.325 


266 


.16754 


.321 


.21759 


.376 


.26998 


.431 


.322 


267 


.16843 


.322 


.21853 


.377 


.27095 


.432 


.32^ 


268 


.16931 


.323 


.21946 


.378 


.27192 


.433 


.32£ 


269 


.17020 


.324 


.22040 


.379 


.27289 


.434 


.326 


270 


.17108 


.325 


.22134 


.380 


.27386 


.435 


.327 


271 


.17197 


.326 


.22227 
G 


.381 


.27483 


.436 


.328 



74 



AREAS OF THE SEGMENTS OF A CIRCLE. 



Table — (Continued). 



Versed 
Sine. 


Seg. Ai-ea. 


Versed 
Sine, 


Seg. Area. 


Versed 
Sine. 


S&g. Area. 


Versed 
Sine. 


Seg. Area, 


,437 


.32986 


.453 


.34576 


.469 


.36171 


.485 


.37770 


,438 


.33085 


.454 


.34676 


.470 


.36271 


.486 


.37870 


.439 


.33185 


.455 


.34775 


.471 


.36371 


.487 


.37970 


.440 


.33284 


.456 


.34875 


.472 


.36471 


.488 


.38070 


.441 


.33383 


.457 


.34975 


.473 


.36571 


.489 


.38169 


.442 


.33482 


.458 


.35074 


.474 


•36671 


.490 


.38269 


.443 


.33582 


.459 


.35174 


.475 


.36770 


.491 


.38369 


.444 


.33681 


.460 


.35274 


.476 


.36870 


.492 


.38469 


. 445 


.33781 


.461 


.35373 


.477 


.36970 


.493 


.38569 


.446 
,447 


.33880 
.33979 


,462 
.463 


.35473 
.35573 


.478 
.479 


.37070 
.37170 


.494 
.495 


.38669 
.38769 


.448 


.84079 


.464 


.35673 


.480 


.37276 


.496 


.38869 


,449 


.34178 


.465 


.35772 


.481 


.37370 


.497 


.38969 


.450 


.34278 


.466 


.35872 


.482 


.37470 


.498 


.39069 


.451 


.34377 


.467 


.35972 


.483 


.37570 


.499 


.39169 


,452 


.34477 


.468 


.36072 


.484 


.37670 


.500 


.39269 



USE OF THE ABOVE TABLE. 

To find the Area of a Segment of a Circle, 
Rule —Divide the height or versed sine by the diameter of tlie cu-cle, and find 
the quotient in the column of versed sines. Take the area noted in the next col- 
umn, and multiply it by the square of the diameter, and it will give the area re- 
quired. 

Example.— Required the area of a segment; its height being 10, and the diame- 
ter of the circle 50 feet. ^ r.<^r. -^ n 
10-r-50=.2, and .2, per table, = .11182 ; then .11182X502 = 2^9.oo Ans. 



LENGTHS OF CIRCULAR ARCS. 



75 



Height. 



Table of the Lengths of Circular Arcs. 

Length. Height. I Length. Height. Length. Height. 



.100 


1.0265 


.156 


1.0637 


.212 


1.1158 


.268 


1.1816 


.101 


1.0270 


.157 


1.0645 


.213 


1.1169 


.269 


1.1829 


.102 


1.0275 


.158 


1.0653 


.214 


1.1180 


.270 


1.1843 


.103 


1.0281 


.159 


1.0661 


.215 


1.1190 


.271 


1.1856 


.104 


1.0286 


.160 


1.0669 


.216 


1.1201 


.272 


1.1869 


.105 


1.0291 


.161 


1.0678 


.217 


1.1212 


.273 


1.1882 


.106 


1.0297 


.162 


1.0686 


.218 


1.1223 


.274 


1.1897 


.107 


1.0303 


.163 


1.0694 


.219 


1.1233 


.275 


1.1908 


.108 


1.0308 


.164 


1.0703 


.220 


1.1245 


.276 


1.1921 


.109 


1.0314 


.165 


1.0711 


.221 


1.1256 


.277 


1.1934 


.110 


1.0320 


.]66 


1.0719 


.222 


J. 1266 


.278 


1.1948 


.111 


1.0325 


.167 


1.0728 


.223 


1.1277 


.279 


1.1961 


.112 


1.0331 


.168 


1.0737 


.224 


1.1289 


.280 


1.1974 


.113 


1.0337 


.169 


1.0745 


.225 


1.1300 


.281 


1.1989 


.114 


1.0343 


.170 


1.0754 


.226 


1.1311 


.282 


1.2001 


.115 


1.0349 


.171 


1.0762 


.227 


1.1322 


.283 


1.2015 


.116 


1.0355 


.172 


1.0771 


.228 


1.1333 


.284 


1.2028 


.117 


1.0361 


.173 


1.0780 


.229 


1.1344 


.285 


1.2042 


.118 


1.0367 


.174 


1.0789 


.230 


1.1356 


.286 


1.2056 


.119 


1.0373 


.175 


1.0798 


.231 


1.1367 


.287 


1.2070 


.120 


1.0380 


.176 


1.0807 


.232 


1.1379 


.288 


1.2083 


.121 


1.0386 


.177 


1.0816 


.233 


1.1390 


.289 


1.2097 


.132 


1.0392 


.178 


1.0825 


.234 


1.1402 


.290 


1.2120 


.123 


1.0399 


.179 


1.0834 


.235 


1.1414 


.291 


1.2124 


,124 


1.0405 


.180 


1.0843 


.236 


1.1425 


.292 


1.2138 


.125 


1.0412 


.181 


1.0852 


.237 


1 . 1436 


.293 


1.2152 


.126 


1.0418 


.182 


1.0861 


.238 


1.1448 


.294 


1.2166 


.127 


1.0425 


.183 


1.0870 


.239 


1.1460 


.295 


1.2179 


.128 


1.0431 


.184 


1.0880 


.240 


1.1471 


.296 


1.2193 


.129 


1.0438 


.185 


1.0889 


.241 


1.1483 


.297 


1.2206 


.130 


1.0445 


.1-86 


1.0898 


.242 


1 . 1495 


.298 


1.2220 


.131 


1.0452 


.187 


1.0908 


.243 


1.1.507 


.299 


1.2235 


.132 


1.0458 


.188 


1.0917 


.244 


1.1519 


.300 


1.2250 


.133 


1.0465 


.189 


1.0927 


.245 


1.1531 


.301 


1.2264 


.134 


1.0472 


.190 


1.0936 


.246 


1.1543 


.302 


1.2278 


.135 


1.0479 


.191 


1.0946 


.247 


1.1555 


.303 


1.2292 


.136 


1.0486 


.192 


1.0956 


.248 


1.1567 


.304 


1.2306 


.137 


1.0493 


.193 


1.0965 


.249 


1.1579 


..305 


1.2321 


.138 


1.0500 


.194 


1.0975 


.250 


1.1591 


.306 


1.2335 


.139 


1.0508 


.195 


1.0985 


.251 


1.1603 


.307 


1.2349 


.140 


1.0515 


.196 


1.0995 


.252 


1.1616 


.308 


1.2364 


.141 


1.0522 


.197 


1.1005 


.253 


1.1628 


.309 


1.2378 


.142 


1.0529 


.198 


1.1015 


.254 


1.1640 


.310 


1.2393 


.143 


1.0537 


.199 


1.1025 


.2.55 


1.1653 


.311 


1.2407 


.144 


1.0544 


.200 


1.1035 


.256 


1.1665 


.312 


1.2422 


145 


1.0552 


.201 


1.1045 


.257 


1.1677 


.313 


1.2436 


.146 


1.0559 


.202 


1.1055 


.258 


1.1690 


.314 


1.2451 


147 


1.0567 


.203 


1.1065 


.259 


1.1702# 


.315 


1.2465 


148 


1.0574 


.204 


1.1075 


.260 


1.1715 


.316 


1.2480 


149 


1.0582 


.205 


1.1085 


.261 


1.1728 


.317 


1.2495 


150 


1.0590 


.206 


1.1096 


.262 


1.1740 


.318 


1.2510 


151 


1.0597 


.207 


1.1006 


.263 


1.1753 


.319 


1.2524 


152 


1.0605 


.208 


1.1117 


.264 


1.1766 


.320 


1.2539 


153 


1.0613 


.209 


1.1127 


.265 


1.1778 


.321 


1.25.54 


154 


1.0621 


.210 


1.1137 


.266 


1.1791 


.322 


1.2569 


.155 


1.0629 1 


.211 1 


1.1148 


.267 


1.1804 1 


.323 


1.2584 



Length. 



76 



LENGTHS OF CIRCULAR ARCS. 



Table— (Continued). 



Length. 



Height. 



Length. 



.2599 


.369 


.2614 


.370 


.2629 


.371 


.2644 


.372 


.2659 


.373 1 


.2674 


.374 1 


.2689 


.375 ! 


.2704 


.376 i 


.2720 


.377 ! 


.2735 


.378 ! 


.2750 


.379 


.2766 


.380 ' 


.2781 


.381 


.2786 


.382 


.2812 


.383 


.2827 


.384 


.2843 


.385 


.2858 


.386 


.2874 


.387 


.2890 


.388 


.2905 


.389 


.2921 


.390 


.2937 


.391 


.2952 


.392 


.2968 


.393 


.2984 


.394 


.3000 


.395 


.3016 


.396 


.3032 


.397 


.3047 


.398 


.3063 


.399 


.3079 


.400 


.3095 


.401 


.3112 


.402 


.3128 


.403 


.3144 


.404 


.3160 


.405 


.3176 


.406 


.3192 


.407 


.3209 


.408 


.3225 


.409 


.,3241 


.410 


.3258 


.411 


L.3274 


.412 


1.3291 





1.3307 
1.3323 
1.3340 
1.3356 
1.3373 
1.3390 
1.3406 
1.3423 
1.3440 
1.3456 
1.3473 
1.3490 
1.3507 
1.3524 
1.3541 
1.3558 
1.3574 
1.3591 
1.3608 
1.3625 
1.3643 
1.3660 
1.3677 
1.3694 
1.3711 
1.3728 
1.3746 
1.3763 
1.3780 
1.3797 
1.3815 
1.3832 
1.3850 
1.3867 
1.3885 
1.3902 
1.3920 
1.3937 
1.3955 
1.3972 
1.3990 
1.4008 
1.4025 
1.4043 



Height. 


Length. 


Height. 


.413 


1.4061 


.457 


.414 


1.4079 


.458 


.415 


1.4097 


.459 


.416 


1.4115 


.460 


.417 


1.4132 


.461 


.418 


1.4150 


.462 


.419 


1.4168 


.463 


.420 


1.4186 


.464 


.421 


1.4204 


.465 


.422 


1.4222 


.466 


.423 


1.4240 


.467 


.424 


1.4258 


.468 


.425 


1.4276 


.469 


.426 


1.4205 


.470 


.427 


1.4313 


.471 


.428 


1.4331 


.472 


.429 


1.4349 


.473 


.430 


1.4367 


.474 


.431 


1.4386 


.475 


.432 


1.4404 


.476 


.433 


1.4422 


.477 


.434 


1.4441 


.478 


.435 


1.4459 


.479 


.436 


1.4477 


.480 


.437 


1.4496 


.481 


438 


1.4514 


.482 


.439 


1.4533 


.483 


.440 


1.4551 


.484 


.441 


1.4570 


.485 


.442 


1.4588 


.486 


.443 


1.4607 


.487 


.444 


1.4626 


.488 


.445 


1.4644 


.489 


.446 


1.4663 


.490 


.447 


1.4682 


.491 


.448 


1.4700 


.492 


.449 


1.4719 


.493 


.450 


1.4738 


.494 


.451 


1.4757 


.495 


.452 


1.4775 


.496 


.453 


1.4794 


.497 


.454 


1.4813 


.498 


.455 


1.4832 


.499 


.456 


1.4851 


.500 



To find tlwLengVi of an Arc of a Circle by the foregoing Table. 

Bt-i f -Divide the heialit by the base, find the quotient in the column of heights, 

the arc required. u ■ „ mn 

ExAMPLE.-What is the length of an arc of a circle, the span or base being 100 

feet, and the height 25 feet "? ,.• v«^ w inn 

25-M00= .25, and .25, per table, gives 1.1591 ; which, being mulUphed by 100, 

= 115.9100, the length. 



LENGTH OF AN ELLIPTIC ARC. 



77 



Note. — When great accuracy is required, if, in the division of a height by the 
base, there should de a remainder. 

Find the lengths of the cuives from the two nearest tabular heights, and sub- 
tract the one length from the other. Then, as the base of the arc of which the 
length is required is to the remainder in the operation of division, so is the differ- 
ence of the lengths of the curves to the complement required, to be added to the 
length. 

Example.— What is the length of an arc of a circle, the base of which is 35 feet 
and the height or versed sine 8 feet 1 * 

8-^35 =.228|5, .228 = 1.1333, .229 = 1.1344, 1.1333X35 = 39.6655, 1.1344X35 
= 39.7040, 39.7040—39.6655 = .0385, difference of lengths. 

Hence, as 35 : 20 : : .0385 : .0220, the length for the remainder, and .0220-f- 
39.6655 = 39.6875, and .6875X12, for inches = 8^ making the length of the arc 39 
feet 84: inches. 



To find the length of an Elliptic Curve which is less than half 
of the entire Figure, 




Geometrically. — Let the curve of which the length is required he aba. 
Extend the versed sine hd to meet the centre of the curve in e. 
Draw the line c e, and from e, with the distance e b, describe bh; bisect c A in e, 
and from e, with the radius e i, describe k i, and it is equal half the arc a be. 

To find the length when the Curve is greater than half the entire 
Figure* 

Rule. — Find by the above problem the curve of the less portion of the figure, 
and subtract it from the circumference of the ellipse, and the remainder will be the 
length of the curve required. 

G2 



78 



LENGTHS OF SEMI-ELLIPTIC ARCS. 



Height. 



Table of the Lengths of Semi-elliptic Arcs. 

Lenstli. Height. ) Length. Height. Length. Height. Length. 



.100 

.101 

.102 

.103 

.104 

.105 

.110 

.115 

.120 

.125 

.130 

.135 

.140 

.145 

.150 

.155 

.160 

.165 

.170 

.175 

.180 

.185 

.190 

.195 

.200 

.205 

.210 

.215 

.220 

.225 

.230 

.235 

.240 

.245 

.250 

.255 

.260 

.265 

.270 

.275 

.280 

.285 

.290 

.295 

.300 

.305 

.310 



1.0416 
1.0426 
1.0436 
1.0446 
1.0456 
1.0466 
1.0516 
1.0567 
1.0618 
1.0669 
1.0720 
1.0773 
1.0825 
1.0879 
1.0933 



_ 0989 
1.1045 
1.1106 
1.1157 
1.1213 
1270 
1327 
1384 
1442 
1501 
1560 
1620 
1680 
1.1741 
1.1802 
1.1864 
1.1926 
1.1989 
1.2051 
1.2114 
1.2177 
1.2241 
1.2306 
1.2371 
1.2436 
1.2501 
1.2567 
1.2634 
1.2700 
1.2767 
1.2834 
1.2901 



315 
320 
325 
330 
,335 
,340 
,345 
.350 
.355 
.360 
.365 
.370 
.375 
.380 
.385 
.390 
.395 
.400 
.405 
.410 
.415 
.420 
.425 
.430 
.435 
.440 
.445 
.450 
.455 
.460 
.465 
.470 
.475 
.480 
.485 
.490 
.495 
.500 
.505 
.510 
.515 
.520 
.525 
.530 
.535 
.540 



1.2960 

1.3038 

1.3106 

1.3175 

1.3244 

1.3313 

1.3383 

1.3454 

1.3525 

1.3597 

1.3669 

1.3741 

1.3815 

1.3888 

1.3961 

1.4034 

1.4107 

1.4180 

1.4253 

1.4327 

1.4402 

1.4476 

1.4552 

1.4627 

1.4702 

1.4778 

1.4854 

1.4931 

1.5008 

1.5084 

1.5161 

1.5238 

1.5316 

1.5394 

1.5472 

1.5550 

1.5629 

1.5709 

1.5785 

1.5863 

1.5941 

1.6019 

1.6097 

1.6175 

1.6253 

1.6331 



545 
550 
555 
560 
,565 
,570 
,575 
.580 
.585 
.590 
.595 
.600 
.605 
.610 
.615 
.620 
.625 
.630 
.635 
.640 
.645 
.650 
.655 
.660 
.665 
.670 
.675 
.680 
.685 
.690 
.695 
.700 
.705 
.710 
.715 
.720 
.725 
.730 
.735 
.740 
.745 
.750 
.755 
.760 
.765 
.770 



1.6409 

1.6488 

1.6567 

1.6646 

1.6725 

1.6804 

1.6S83 

1.6963 

1.7042 

1.7123 

1.7203 

1.7283 

1.7364 

1 . 7444 

1.7525 

1.7606 

1.7687 

1.7768 

1.7850 

1.7931 

1.8013 

1.8094 

1.8176 

1.8258 

1.8340 

1.8423 

1.8505 

1.8587 

1.8670 

1.8753 

1.8836 

1.8919 

1.9002 

1.9085 

1.9169 

1.9253 

1.9337 

1.9422 

1.9506 

1.9599 

1.9675 

1.9760 

1.9845 

1.9931 

2.0016 

2.0102 



775 

780 
785 
790 
795 
,800 
,805 
,810 
.815 
.820 
.825 
.8.30 
.835 
.840 
.845 
.850 
.855 
.860 
.865 
.870 
.875 
.880 
.885 
.890 
.895 
.900 
.905 
.910 
.915 
.920 
.925 
.930 
.935 
.940 
.945 
.950 
.955 
.960 
.965 
.970 
.975 
.980 
.985 
.990 
.995 
.1000 



2.0187 
2.0273 
2.0360 
2.0446 
2.0533 
2.0620 
2.0708 
2.0795 
2.0883 
2.0971 
2.1060 
2.1148 
2.123T 
2.1326 
2.1416 
2.1505 
1595 
1685 
1775 
1866 
1956 
2.2047 
2.2139 
2.2230 
2.2322 
2.2414 
2.2506 
2.2597 
2.2689 
2.2780 
2.2872 
2.2964 
2.3056 
2.3148 
2.3241 
2.3335 
2.3429 
2.3524 
2.3619 
2.3714 
2.3810 
2.3906 
2.4002 
2.4098 
2.4194 
2.4291 
I _^ 



To find the Length of the Curve of a Right Semi-Ellipse. 

Proceed with the foregoing table by the rules for ascertaining the lengths of cir- 
cular arcs, page 76. 

Example.— What is the length of the curve of the arch of a bridge, the spam 
being 70 feet, and the height 30.10 feet 1 

30.10^70 =.430 = per table, 1.4627, and 1.4627X70 = 102.3890, the length re 
quired. 



SEMI-ELLIPTIC ARCS. 79 

When the Curve is not that of a Right Semi-Ellipse^ the height 
being half of the Transverse Diameter. 

Rule. — Divide half the base by twice the height ; then proceed as in the forego- 
ing example, and multiply the tabular length by twice the height, and the product 
will be the length required. 

Example.— What is the length of the profile of arch (it being that of a semi-el- 
lipse), the height measuring 35 feet and the base 60 feet 1 

60-^2=1:30-7 -35x2 = .428, the tabular length of which is 1.4597. 
Then, 1.4597x35X2=102.1790, the length required. 
Note. — When the quotient is not given in the column of heights, divide the dif- 
ference between the two nearest heights by .5 ; multiply the quotient by the excess of 
the height given and the height in the table first above it, and add this sum to the 
tabular area of the least height. Thus, if the height is 118, 
.115, per table, = 1.0567 
.120, " = 1.0618 

.0051-S-.5 = .00102 X (118 — 115) = .00306, 
which, added to 1.0567 = 1.05976, the length for 118. 



80 



AREAS OF THE ZONES OF A CIRCLE. 



Table of the Areas of the Zo7ies of a Circle. 



Height. 


Area. 


Height. 


Area. i 


Height. 


Area. 


Height. 


Area. 


.001 


.00100 


.115 


.11397 


.245 


.23480 


.375 


.33604 


.002 


.00300 


.120 


.11883 


.250 


.23915 


.380 


.33931 


.003 


.00300 


.125 


.12368 


.255 


.24346 


.385 


.34253 


.004 


.00400 


.130 


.12852 


.260 


.24775 


.390 


.34569 


.005 


.00500 


.135 


.13334 


.265 


.25201 


.395 


.34879 


.010 


.01000 


.140 


.13814 


.270 


.25624 


.400 


.35182 


.015 


.01499 


.145 


.14294 


.275 


.26043 


.405 


.35479 


.020 


.01999 


.150 


.14772 


.280 


.26459 


.410 


.35769 


.025 


.02499 


.155 


.15248 


.285 


.26871 


.415 


.36051 


.030 


.02998 


.160 


.15722 


.290 


.27280 


.420 


.36326 


.035 


.03497 


.165 


.16195 


.295 


.27686 


.425 


.36594 


.040 


.03995 


.170 


.16667 


.300 


.28088 


.430 


.36853 


.045 


.04494 


.175 


.17136 


.305 


.28486 


.435 


.37104 


.050 


.04992 


.180 


.17603 


.310 


.28880 


.440 


•37346 


.055 


.05489 


.185 


.18069 


.315 


.29270 


.445 


.37579 


.060 


.05985 


.190 


.18532 


.320 


.29657 


.450 


.37805 


.065 


.06482 


.195 


.18994 


.325 


.30039 


.455 


.38015 


.070 


.06977 


.200 


.19453 


.330 


.30416 


.460 


.38216 


.075 


.07472 


.205 


.19910 


.335 


.30790 


.465 


.38466 


.080 


.07965 


.210 


.20365 


.340 


.31159 


.470 


.38853 


.085 


.08458 


.215 


.20818 


.345 


.31523 


.475 


..38747 


.090 


.08951 


.220 


.21268 


.350 


.31883 


.480 


.38895 


.095 


.09442 


.225 


.21715 


.355 


.32237 


.485 


.39026 


.100 


.09933 


.230 


.22161 


.360 


.32587 


.490 


.39137 


.105 


.10422 


.235 


.22603 


.365 


.32931 


.495 


.39223 


.110 


.10910 


.240 


.23t)43 


.370 


.33270 


.500' 


.39270 



To find the Area of a Zone hy the above Tahle. 

Rule 1. — When the zone is greater than a part of a semicircle, take the height 
on each side of the diameter of the circle, of which it is a part; divide the heights 
by the diameter ; find the respective quotients in the column of heights, and take 
out the areas oppot^ite to them, multiplying the areas thus found by the square of 
the diameter or chord, and the products, added together, will be the area required. 

Note. — When the quotient is not given in the column of heights, divide the differ- 
ence between the two nearest heights by 5, and multiply the quotient by the excess be- 
tween the height giver^and the height in the table first above it, and add this sum to 
the tabular area of the least height. Thus, if the height is .333, 

.30416— .30790= .00374-^-5= .000748x3 (excess of 333 over 330) = .002244+.30416 
= .306404, the area for 333. 

Example. — What is the area of zone, the diameter of the circle being 100, and 
the heights respectively 20 and 10, upon each side of it ? 

20-1-100 = .200, and 200, per table, = .19453x1002 = 1945.3. 
10-J-lOO = .100, and 100, per table, = .09933X 100^ = 993.3. 
Hence, 1945.3+993.3 = 2938.6 ^ns. 



RULE.- 

height. 



'When the zone is less than a semicircle, proceed as in rule 1 for one 



Example. — What is the area of a zone, the longest chord being 10, and the 
height 4 ? 

4-r-lO = .400 = .35182X 10^ = 35.182 .^ns. 



MENSURATION OF SOLIDS. 



81 



MENSURATION OF SOLIDS. 

23. OF CUBES AND PARALLELOPIPEDONS. 

24. 





To find the Solidity of a Cube— fig. 23, 
Rule. — Multiply the side of the cube by itself, and that product 
again by the side, and this last product will be the solidity. 

To find the Solidity of a ParaUelopipedon—fig. 24. 
Rule. — Multiply the length by the breadth, and that product by 
the depth, and this product is the solidity. 

OF REGULAR BODIES. 
To find the Solidity of any Regular Body. 
Rule. — Multiply the tabular solidity in the following table by the 
cube of the linear edge, and the product is the solidity. 

Table of the Solidities of the Regular Bodies when the Lirwar Edge is 



Number of Sides, 


Names. 


Solidities. 


4 


Tetrahedron. 


0.11785 


6 


Hexahedron. 


1. 00000 


8 


Octahedron. 


0.47140 


12 


Dodecahedron. 


7.66312 


20 


Icosahedron. 


2.18169 



OF CYLINDERS, PRISMS, AND UNGULAS. 
25- 28. ^ 27 





To find the Solidity of Cylinders^ Prisms, and Ungulas—figs. 25, 26, 

aTid 27. 
Rule. — Multiply the area of the base by the height, and the prod- 
uct is the solidity. 



82 MENSURATION OF SOLIDS. 

To find the Solidity of an Ungula^fig. 28, when the section passes 
obliqvAy through the cylinder^ abed. 

Rule.— Multiply the area of the base of the cylinder by half the 
sum of the greater and less heights a e, of of the ungula, and the 
product is the solidity. 

When the Section passes through the base of the Cylinder and one of its 
sides— fig. 29, a be. 

Rule.— Frorii | of the cube of the right sine a d, of half the arc 
ag of the base, subtract the product of the area of the base, and the 
cosine df of said half arc. Multiply the difference thus found by 
the quotient of the height, divided by the versed sine, and the prod- 
uct is the solidity. 

Whe7i the S2ction passes ooliquely through both ends of the Cylinder^ 
adc e—fig. 30. 

Rule.— Find the solidities of theungulas adce and dbc, and the 
difference is the solidity required {conceiving the section to be con- 
tinued till it meets the side of the cylinder). 

Note.— For rules to ascertain the solidity of conical ungulas, see RyarCs Bonny- 
castle's Mensuration^ page 136 (1839). 

OF CONES AND PYRAMIDS. 
33. 




c a 

To find the Solidity of a Cone or Pyramid—figs. 31 and 33. 

Rule.— Multiply the area of the base by the height c d, and I the 
product vi^ill be the content. 

To find the Solidity of the Frustrum of a Cone— fig. 32. 

Rule. — Divide the difference of the cubes of the diameters ah.cd 
of the two ends by the difference of the diameters ; this quotient, 
multiplied.by .7854, and again by J of the height, will give the so- 
lidity. 

To find tlie Solidity of the Frustrum of a Pyramid— fig. 34. 

Rule.— Add to the areas of the two ends of the frustrum the 
square root of their product, and this sum, multiplied by J of the 
height a b, will give the solidity. 



MENSURATION OF SOLIDS. 



83 



OF WEDGES AND PRISMOIDS. 
d 




To find the Solidity of a Wedge— fig. 35. 

Rule.— To the length of the edge of the wedge de add twice the 
length of the back a h ; multiply this sum by the height of the wedge 
df and then by the breadth of the back c a, and i of the product 
will be the solid content. 

To find the Solidity of a Prismoid—fig. 36. 

Rule.— Add the areas of the two ends ahc, def and four times 
the middle section g h, parallel to them, together ; multiply this sum 
by ^ of the height, and it will give the solidity. 



OF SPHERES. 

c 




To find the Solidity of a Sphere— fig. 37. 

Rule —Multiply the cube of the diameter by .5236, and the prod- 
uct IS the solidity. ^ 

To find th£ Solidity of a Spherical Segment— fig. 38. 
Rule.— To three times the square of the radius of its base a b, add 
the square of its height cb; then multiply this sum by the height 
and the product by .5236. ^ 

To find the Solidity of a Spherical Zone or Frustrum—fig. 39. 
Rule.—To the sum of the squares of the radius of each end 
ab.cd, add I of the square of the height b d of the zone ; and this 
th"^' rd^^^ ^^ ^^^ ^^^^^^' ^"^ ^^^ product by 1.5708, will give 



84. 



MENSTJRATION OF SOLIDS. 



OF SPHEROIDS 




To find the Solidity of a Spheroid— fig. 40. 
Rule.— Multiply the square of the revolving axis cdhy the fixed 
axis ah; the product, multiplied by .5236, will give the solidity. 

To find the Solidity of the Segment of a Spheroid— figs, 41 and 42, 

I^uLE. When the base efts circular, or parallel to the revolving axis 

cd, fig. 41. Multiply the fixed atis a^> by 3, the height of the seg- 
ment^^a g by 2, and subtract the one product from the other ; then 
multiply the rem.ainder by the square of the height of the segment, 
and the product by .5236. Then, as the square of the fixed axis is 
to the square of the revolving axis, so is the last product to the 
content of the segment. 

Rvh^.—When the base ef is perpendicular to the revolving axis cdy 
fig. 42. Multiply the revolving axis by 3, and the height of the seg- 
ment c^ by 2, and subtract the one from the other; then multiply 
the rentainder by the square of the height of the segment, and the 
product by .5236. Then, as the revolving axis is to the fixed axis, 
so is the last product to the content. 

To find the Solidity of the Middle Frustrum of a Spheroid— figs, 43 

and 44. 

Rule.— WAcTi the ends ef and gh are circular, or parallel to the re- 
volving axis c d, fig. 43. To twice the square of the revolving axis c d, 
add the square of the diameter of either end, ef or g h ; then multi- 
ply this sum by the length a ^ of the frustrum, and the product again i 
by .2618, and this will give the solidity. 

Rule.— T7/ien the ends ef and gh are elliptical or perpendicular to 
the revolving axis c d, fig. 44. To twice the product of the transverse 
and conjugate diameters of the middle section ab, add the product 
of the transverse and conjugate of either end ; multiply this sum by 
the length Ik of the frustrum, and the product by .2618, and this will 
give the solidity. 

* Spheroids are either Prolate or Oblate. They are prolate when produced by th 
revolution of a semi-ellipse about its transverse diameter, and oblate when produc( 
by an ellipse revolving about its conjugate diameter. 



MENSURATION OF SOLIDS. 



85 




OF CIRCULAR SPINDLES. 
46. 



\ ! 



To find the Solidity of a Circular S'pindle—fig. 45. 
Rule.— Multiply the central distance oe by half the area of the 
revolving segment a c ef. Subtract the product from J of the cube 
fe of half the length ; then multiply the remainder by 12.5664 (or 
four times 3.1416), and the product is the solidity. 

To find the Solidity of the Frustrum, or Zone of a Circular Spindle — 
fig' 46. 
Rule.— From the square of half the length h i of the whole spin- 
dle, take I of the square of half the length n i of the frustrum, and 
multiply the remainder by the said half length of the frustrum ; mul- 
tiply the central distance o i by the revolving area* which generates 
the frustrum ; subtract the last product from the former, and the 
remainder, multiplied by 6.2832 (or twice 3.1416), will give the so- 
lidity. 

ELLIPTIC SPINDLES. 

48, 








To find the Solidity of an Elliptic Spindle— fig. 47. 
Rule. — To the square of the greatest diameter a b, add the square 
of twice the diameter ef at i of its length ; multiply the sum by the 
length, and the product by .1309, and it will give the solidity nearly. 

To find the Solidity of a Frustrum or Segment of an Elliptic Spindle — 

fig. 48. 
Rule. — Proceed as in the last rule for this or any other solid 
formed by the revolution of a conic section about an axis, viz. : Add 
together the squares of the greatest and least diameters, ab.cd, and 
the square of double the diameter in the middle, between the two ; 
multiply the sum by the length ef and the product by .1309, and it 
will give the solidity. 

Note. — For all such solids, this rule is exact when the body is formed by the conic 
•action, or a part of it, revolving about the axis of the section, and will always be 
very near when the figure revolves about another line. 

* The area of the frustrum can be obtained by dividing its central plane into seg- 
ments of a circle, and triangles or parallelograms. 

H 



86 



MENSURATION OF SOLIDS. 



OF PARABOLIC CONOIDS AND SPINDLES. 
J? 50. 





a!^-—^i. 



To find tlie Solidity of a Paraholic Co7wid*—fig. 49. 
Rule.— Multiply the area of the base dchy half the altitude fg, 
and the product will be the solidity. 

Note.— This rule will hold for any seg-ment of the paraboloid, whether the base 
be perpendicular or oblique to the axis of the solid. 

To find the Solidity of a Frustrum of a Paraboloid— fig. 49. 
Rule. — Multiply the sum of the squares of the diameters ah and 
dchy the height ef, and the product by .3927. 

To find the Solidity of a Parabolic Spindle— fig. 50. 
Rule. — Multiply the square of the diameter ah hy the length rfc, 
and the product by .4188, and it will give the solidity. 

To find the Solidity of the Middle Frusf.rum of a Parabolic Spindle— 

fig- 51. 
Rule.— Add together 8 times the square of the greatest diameter- 
c<Z, 3 times the Square of the least diameter e/, and 4 times the' 
product of these two diameters ; multiply the sum by the length a h, . 
and the product by .05236, and it will give the solidity. 



OF HYPERBOLOIDS AND HYPERBOLIC CONOIDS. 

c 

53. ^-'^^^^ 

1 i^ 



To find the Solidity of a Hyperholoid—fig. 52. 

Rule.— To the square of the radius of the base a b, add the square 

of the middle diameter n m ; multiply this sum by the height c r, and i 

the product again by .5236, and it will give the solidity. 

* The parabolic conoid is := ^ its circumscribing cylinder. 




MENSURATION OF SOLIDS. 87 

To find the Solidity of the Frustrum of a Hyperbolic CoTwid—fig. 53. 
Rule. — Add together the squares of the greatest and least semi- 
diameters a s and d r, and the square of the whole diameter nm in 
the middle of the two : multiply this sum by the height r s, and the 
product by .5236, and -it will give the solidity. 

OF CYLINDRICAL RINGS. 




To find the Solidity of a Cylindrical Ring-^fig. 54. 

Rule. — To the thickness of the ring a b, add the inner diameter 
he; then multiply the sum by the square of the thickness, and the 
product by 2.4674, and it will give the solid.ity. 



BY MATHEMATICAL FORMULA. 

FRUSTRUM OF A RIGHT TRIANGULAR PRISM. 

The base Xi[h-{-h'-]-h"), h being the heights. 

FRUSTRUM OF ANY RIGHT PRISM. 

The base X its distance from the centre of gravity of the section. 

CYLINDRICAL SEGMENT. 

Contained between the base and an oblique plane passing through 
a diameter of the base, twice the height x the quotient of the square 
of the radius -:-3 ; or f Ar^ r being the radius and h the height. 

SPHERICAL SEGMENT. 

— (Sr'^+A^), r being the radius of the base, and h the height of the 
segment. 

SPHERICAL ZONE. 

^(3R--f 3r2+A2^, Rr being the radii of the bases. 

SPHERIQAL SECTOR. 

Jr X the surface of the segment or zone. 

2 ELLIPSOID. 

^-^— , a being the revolving diameter, and b the axis of revolution. 
6 



88 MENSURATION OF ScJlIDS. 

PARABOLOID. 

i area of the base X the height. 

CIRCULAR SPINDLE. 



p[^c^ — 'Hsy/r^ — \c^), s being the area of the revolving segment, 
and c its chord. 

ANY SOLID OF REVOLUTION. 

2prs ; or, the area of the generating surface X the circumference 
described by its centre of gravity. 

Note. — If bounded by a curved surface, find the area by the rule for irregular 
plane figures. 

To find a Cylinder of a Given Solidity (b) with the Least Surface, 
Let a = altitude, and d = diameter of base. 

Then = h, and ^— == sum of bases. 

4 2 

Convex surface . =pday or — . 

Therefore . =^ + — rr= a mmimum. 
3 a 



CASK GAUGING. 

Casks are usually comprised under the following figures, viz. : 

1. The middle frustrum of a spheroid. 

2. The middle frustrum of a parabolic spindle. 

3. The two equal frustrums of a paraboloid. 

4. The two equal frustrums of a cone, 

and their contents can be computed by the preceding rules for these 
figures. 

To find the Content of a Cask by four Dimensions. 

Rule. — x\dd together the squares of the bung and head diame- 
ters, and the square of double the diameter taken in the middle be- 
tween the bung and head ; multiply the sum by the length of the 
cask, and the product by .1309. 

Or, /.1309 (^2_|_]32^2M2), / being the length, dJ) the head and 
bung diameters, and M a diameter midway between them. 

Or, /( q^^)' ^DM being the areas of the diameters. 

^ , ^2D + 6Z 

Or, I X area of — - — . 

o 

ULLAGE CASKS. 
Wlien the Cask is standins[. 



/ x( ^ ), I being the height of the fluid. 



MENStTRATION OF SOLIDS. 89 

When the Cask is on its bilge. 

Rule. — Divide the wet inches by the bung diameter, and oppo- 
site to the quotient in the column ox versed sines, page 72, take the 
area ^f the segment ; multiply this area by the content of the cask 
in inches or gallons, and the product by 1.25 for the ullage required. 

Example.— What is the ullage of a cask that contains 25689 
cubic inches, and has 8 inches of liquor on its bilge, the bung diam- 
eter 32 inches 1 Ans. 4930. 



Examples in Illustration of the foregoing Rules. 

1. The side of a cube abc, fig. 23, is 5 inches ; what is the solidity 1 

Jlns. 125. 

2. The length of a parallelopipedon a h, fig. 24, is 8 inches, its depth h c and 
breadth cdi inches ; what is the solidity 1 Ans. 128. 

3. The base of a cylinder a b, fig. 25, is 30 inches, and the height 6 c 50 inches -^ 
what is the solidity f ^ns. 20.4531 cubic feet. 

4. The sides of a prism ab, be, fig. 26, are each 5 inches, and the height bd 10 
inches ; what is the solidity 1 

5+5+5 

■ ^ =7.5; then 7.5—5 . 7.5-5 . 7.5— 5 = 2.5 .v/7.5X2.5x2.5x2.5Xl0 = 

108.6 .^ns. 

5. The base of an ungula, fig. 27, is a semicircle, the radius 5 inches, and the 
height of the figure 25 inches ; what is the solidity ? Ans. 981.75. 

6. The base of an ungula, fig. 28, is a circle, the radius 8 inches, and the heights 
of the sides 2 and 4 inches; what is the solidity 1 Ans. 603.180. 

7. The base of g, cone, fig. 31, is 20 inches, and the. height c d 24 ; what is the 
solidity 1 Ans. 2513.28. 

8. The diameter of the greater end cd of the frustrum of a cone, fig. 32, is 5 feet, 
the less end ab 3 feet, and the perpendicular height 9 feet ; what is the solidity 1 

Ans. 115.4538. 

9. What is the solidity of the frustrum of a hexagonal pyramid, fig. 34, a side c d 
of the greater end being 4 feet, and one of the lesser end 3 feet, and the height ab 
9 feet 1 Ans. 288.3864. 

10. How many solid feet are there in a wedge, fig. 35, the base aft is 64 inches 
long, c a 9 inches, de4:2 inches, and the height df 28 inches 1 Ans. 4.1319. 

11. What is the solidity of a rectangular prismoid, fig. 36, the base being 12 by 
14 inches, the top 4 by 6 inches, and the perpendicular height 18 feet 1 

Ans. 10.666. 

12. What is the solidity of the sphere, fig. 37, the diameter being 17 inches 1 

Ans. 2572.4468. 

13. The segment of a sphere, fig. 38, has a radius ab oil inches for its base, and 
its height 6 c is 4 inches ; what is the solidity 7 Ans. 341.3872. 

14. The greater and less diameters of a spherical zone, fig. 39, are each 3 feet, 
and the height db A feet ; what is the solidity in cubic feet ? ^715. 61.7843. 

15. In the prolate spheroid, fig. 40, the fixed axis a 6 is 100, and the revolving 
axis c d is 60 ; what is the solidity ? Ans. 188496. 

16. The segment of a spheroid, fig. 41, is cut from a spheroid, the fixed axis of 
which, ab, is 100 inches, the height of the segment ag is 10, and the revolving 
axis 60 ; what is the solidity of it ? Ans. 14660.8. 

17. The fixed axis cc?, fig. 42, is 60, the revolving axis 100, and the height of the 
segment cgQ inches ; what is its solidity 1 Ans. 9159.509. 

18. What is the solidity of the middle frustrum of a spheroid, fig. 43, the middle 
diameter c d being 60, ef and g h each 36, and the length a 6 801 

Atis. 17?M0.224. 
H2 



90 MENSUEATION OF SOLIDS. 

19. In the middle fnistrum efgh, fig. 44, an ohlate spheroid, the diameters of (he 
ends are 40 by 24 inches, the middle diameters are 50 and 30, and the height Ik is 
18 inches ; what is the solidity 7 jins. ]8661.104. 

20. What is the solidity of a circular spindle, fig. 45, the distance o e 7.07 inches, 
the length /e 7.07 inches, and the radius oc 10 inches'? Ans. 210.96. 

21. The length of an elliptic spindle ah, fig. 47, is 85, its greatest diameter c d 25, 
and the diameter e/, at ^ of its length, 20 inches ; what is its solidity ? 

Ans. 2475G.4625. 

22. What is the solidity of the parabolic conoid, fig. 49 ; its height gfis 60, and 
the diameter dc of its base 100 inches 1 jins. 235620. 

23. What is the solidity of the parabolic frustrum abed, fig. 49, its diameters de 
58, a b 30, and height ef 18 inches ? ^ns. 30140.5104. 

24. What is the solidity of the parabolic spindle, fig. 50, a b being 40, and c d 100 
laches 7 ^ns. 67008.8. 

25. The middle frustrum of a parabolic spindle, fig. 51, has ab 60, cd 40, and ef 
30 inches ; what is its solidity 1 jjns. 63774.48. 

26. In the hyperboloid, fig. 52, the height c r is 10, the radius a r 12, and the 
middle diameter n m 15.8745 ; what is the solidity 1 Ans. 2073.454691. 

27. The frustrum of the hyperbolic conoid, fig. 53, has rs l%ablQ,de 6, and 
nm 8.5 inches ; what is the solidity 7 jins. 667.59. 

28. The thickness ab, fig. 54, of a cylindric ring is 3 inches, and the inner diame- 
ter c 6 8 inches ; what is the solidity 7 Ans. 244.2726. 

29. Required the solidity of an icosahedron, its linear edge being 2 7 

Ans. 17.45352. 

30. Required the content of a cask, the length being 40, the bung and head di- 
ameters 24 and 32, and the middle diameter 28.75 inches. Ans. 25689.125. 



AREAS OF CIRCLES. 



Areas of Circles, from 1 to 100. 



Diametei 


Area. 


Diameter. 


Area. 


Diameter 


Area. 


1 Diameter. 


Area 


1 

■5-5- 


.00019 


5. 


19.635 


12. 


113.09 


19. 


283 


32 


.00076 


1 

•8 

4 


20.629 
21.647 


•i 

4 


115.46 
117.85 


:l 


287 
291 


rV 


.00306 


•8 


22.690 


4 


120.27 


3 
.3 


294 


1 

■g- 


.01227 




23.758 


•t 


122.71 


.i 


298 


3 


.02761 


•1 


24.850 




125.18 




302 


TF 


•? 


25.967 


• 1 


127.67 


'I 


306 


1 

4 


.04908 


.1 


27.108 


7 

•8 


130.19 


.| 


310 


3 


.07669 


6. 


28.274 


13. 


132.73 


20.' 


314. 




•i 


29.464 


•f 


135.29 


.1 


318 


t 


.1104- 


• i 


30.679 




137.88 


•5 


322 


iV 


.1503 




31.919 


• i 


140.50 


•f 


326. 


1 




• ^ 


33.183 


• r 


143.13 




330. 


2 


.1963 


.1 


34.471 


•8 


145.80 


[1 


334. 


9 

TF 


.2485 


•f 


35.784 


• 5 


148.48 


[I 


338. 


5 




4 


37.122 


.1 


151.20 


!l 


342. 


Y 


.3067 


7. 


38.484 


14. 


153.93 


21!' 


346. 


H 


.3712 


.■• 


39.871 


.1 


156.69 


.| 


350. 


3 


.4417 




41.282 


1 

• 4- 


159.48 


'i 


354. 


1 3 


•8 


42.718 




162.29 


* 
.# 


358. 


IT 


.5184 


• 2 


44.178 


.¥ 


165.13 


• Y 


363. 


7 
•5- 


.6013 


_5 


45.663 


,| 


167.98 


.1 


367. 


15 


.6902 


•1 


47.173 


.5 


170.87 . 


.5 


371. 


TF 


•i 


48.707 


*| 


173.78 


7 


375. 


. 


.7853 


8. 


50.265 


15.' 


176.71 


22." 


380. 


•i 


.9940 


■ i 


51.848 


A 


179.67 


.1 


384. 


4 


1.227 


•? 


53.456 


.i 


182.65 


a* 


388. 


•1 


1.484 


•8 


55.088 


J 


185.66 


.f 


393. 


• i 


1.767 


• i 


56.745 


.i 


188.69 




397. 


• 1 


2.073 




58.426 


.1 


191.74 


[i 


402. 


4 


2.405 




60.132 


A 


194.82 


*l 


406. 


7 


2.761 


•f 


61.862 


7 
.1 


197.93 


4 


410. 


1 


3.141 


9. 


63.617 


16. 


201.06 


23. 


415. 


'"i 


3.546 


• I 


65.396 




204.21 




420. 


a 


3.976 


• i 


67.200 


•! 


207.39 


!l 


424. 


• f 


4.430 


•8 


69.029 


.1 


210.59 


_ 


429. 


• i 


4.908 


• i 


70.882 


.i 


213.82 


^ — 


433. 


• 1 


5.411 


• 1 


72.759 


.1 


217.07 


^1 


438. 


• 5 


5.939 


.1 


74.662 


220.35 


[3 


443. 


•1 


6.491 


•8 


76.588 


.8 


223.65 


. g 


447. 


t. 


7.068 


10. 


78.539 


17. 


226.98 


24. 


452. 


• s 


7.669 


4 


80.515 


.1 


230.33 




457. 


• i 


8.295 


1 

• 4 


82.516 


.i 


233.70 


1 


461. 


• ? 


8.946 


• 1 


84.540 


.¥ 


237.10 


^_ 


466. 


• i 


9.621 


.4 


86.590 


.i 


240.52 


i. 


471.^ 




10.320 


^ 


88.664 


.1 


243.97 




476. 




11.044 


•5 


90.762 


A 


247.45 


3 


481. 


•1 


11.793 


•i 


92.885 




250.94 


.1 


485. < 


L 


12.566 


11. 


95.033 


\%\\ 


254.46 


25.' 


490.^ 


•^ 


13.364 


•8 


97.205 




258.01 


• i 


495.' 


• i 


14.186 


•5 


99.402 




261.58 




500.' 


•8 


15.033 


. i 


101.62 




265.18 


'f 


505.' 


• i 


15.904 


•^ 


103.86 


•t 


268.80 


[^ 


510.' 


• ? 


16.800 


• 8 


106.13 




272.44 


• 1 


515.' 


• i 


17.720 


, 1 


108.43 




276.11 


• £ 


520.' 


• 7 


18.665 


•1 


110.75 


ii 


279.81 


ll 


525. i 



92 



AREAS OF CIRCLES. 



Table — (Continued). 



Diameter 


Area. 


Diameter. 


Area. Diam< 


Act. 


Area. Diame 


ter. 


Area. 


26. 


530.93 


33. 


855.30 40 




1256.6 47. 




1734.9 




536.04 


.1 


861.79 


1 


1264.5 


i 


1744.1 


'1 


541.18 


'i 


868.30 


4 


1272.3 


i 


1753.4 


[i 


546.35 


•¥ 


874.84 


3 
8 


1280. 3 


f 


1762.7 


.*! 


551.54 


" -T 


881.41 




1288.2 


^ 


1772.0 


• 2 

.1 


556.76 


5 

• '8 


888.00 


1 


1296.2 


1 


1781.3 


• s 


562.00 




894.61 


I 


1304.2 


i 


1790.7 


-1 


567.26 


♦1 


901.25 


7 
8 


1312.2 


1 


1800.1 


27/ 


572.55 


34. 


907.92 41 




1320.2 48 




1809.5 




577.87 


'i 


914.61 


1 


1328.3 


1 

8 


1818.9 


583.20 


'i 


921.32 


i 


1336.4 


i 


1828.4 


588.57 




928.06 


i 


1344.5 




1837.9 


•- 


593.95 


i 


934.82 


i 


1352.6 


^ 


1847.4 


599.37 


•f 


941.60 


t 


1360.8 


5 
8 


1856.9 


604.80 




948.41 


I 


1369.0 


1 


1866.5 


28*/ 


610.26 


. 7 


955.25 


i 


1377.2 


i 


1876.1 


615.75 


35. 


962.11 42 




1385.4 49 




1885.7 




621.26 


'i 


968.99 


i 


1393.7 


i 


1895.3 


626.79 


4 


975.90 


i 


1401.9 




1905.0 


.1 


632.35 


3 
•8 


982.84 


i 


1410.2 


1 


1914.7 


637.94 




989.80 


i 


1418.6 


^ 


1924.4 


643.54 


•1 


996.78 


1 


1426.9 




1934.1 


649.18 


. .2 


1003.7 


1 


1435.3 




1943.9 


4 


654.83 


• i 


1010.8 


i 


1443.7 


8 


1953.6 


29/ 


660.52 


36. 


1017.8 43 




1452.2 50 




1963.5 




666.22 




1024.9 


I 


1460.6 


^ 


1973.3 


671. t)5 


» z 


1032.0 


I 


1469.1 


^ 


1983.1 


677.71 


• 1 


1039.1 


f 


1477.6 


f 


1993.0 


683.49 


• i 


1046.3 


i 


1486.1 


^ 


2002.9 


689.29 


^ 


1053.5 


1 


1494.7 


•I 


2012.8 


695.12 


. 4 


1060.7 


1 


1503.3 


,| 


2022.8 


30]^ 


700.98 


7 
. 8 


1067.9 


i 


1511.9 


7 
•8 


2032.8 


706.86 


37. 


1075.2 44 




1520.5 51 




2042.8 




712.76 


'i 


1082.4 


_ 


1529.1 


"l 


2052.8 


718,69 


? 


1089.7 


_ 


1537.8 


•i 


2062.9 


11 


724.64 




1097.1 


:i. 


1546.5 


3 

•5 


2072.9 


• i 


730.61 


4 


1104.4 


^ 


1555.2 


•i 


2083.0 


•1 


736.61 


J 


1111.8 




1564.0 


•3 


2093.2 


• 3 


742.64 


•1 


1119.2 


5 


1572.8 


.1 


2103.3 


•1 


748.69 


•J 


1126.6 


i 


1581.6 


7 


2113.5 


3l/ 


7.54.76 


38/ 


1134.1 45 




1590.4 52 




2123.7 


• ^ 


760.86 


,_ 


1141.5 


i 


1599.2 


;• 


2133.9 




766.99 


. 5 


1149.0 


1 

4- 


1608.1 


;: 


2144.1 




773.14 


•t 


1156.6 


1 


1617.0 


r. 


2154.4 


^1. 


779.31 


'k 


1164.1 


i 


1625.9 




2164.7 




785.51 


5 

. 8 


1171.7 


1 


1634.9 


• 1 


2175.0 


] 


791.73 


.1 


1179.3 




1643.8 




2185.4 


^: , 


797.97 


7 

. a 


1186.9 


7 
8 


1652.8 


F 


2195.7 


32*. 


804.24 


39.' 


1194.5 46 




1661.9 53 




2206.1 


.? 


810.54 


•i 


1202.2 


'•i 


1670.9 


4 


2216.6 


• i 


816.86 


.i 


1209.9 


.i 


1680.0 


.i 


2227.0 




823.21 




1217.6 


n 

5 


16S9.1 


.1 


2237.5 


• ^ 


829.57 


• i 


1225.4 


•t 


1698.2 


A 


2248.0 


• 8' 


835.97 


.1 


1233.1 




1707.3 


.1 


2258.5 




842.30 


. i 


1240.9 


.? 


1716.5 


.t 


2269.0 


:l 


848.83 


7 
•I 


1248.7 


.{ 


1726.7 


7 
.1 


2279.6 



AREAS OF CIRCLES. 



Table — (Continued). 



■•I 



2290.2 


61. 


2922.4 


2300.8 


.1 


2934.4 


2311.4 


'1 


2946.4 


2322.1 


•3 


2958.5 


2332.8 


•i 


2970.5 


2343.5 




2982.6 


2354.2 


•1 


2994.7 


2365.0 


7 


3006.9 


2375.8 


62. 


3019.0 


2386.6 


•1 


3031.2 


2397.4 


.1 


3043.4 


2408.3 


•f 


3055.7 


2419.2 


•i 


3067.9 


2430.1 


5 

. 8 


3080.2 


2441.0 


'I 


3092.5 


2452.0 


7 


3104.8 


2463.0 


63.' 


3117.2 


2474.0 




3129.6 


2485.0 


•i 


3142.0 


2496.1 


• ¥ 


3154.4 


2507.1 


•f 


3166.9 


2518.2 




3179.4 


2529.4 


• 1 


3191.9 


2540.5 


• 8 


3204.4 


2551.7 


64. 


3216.9 


2562.9 


'i 


3229.5 


2574.1 


.i 


3242 . 1 


2585.4 


.^ 


3254.8 


2596.7 


.| 


3267.4 


2608.0 




3280.1 


2619.3 


•1 


3292.8 


2630.7 


7 
• 8 


3305.5 


2642.0 


65. 


3318.3 


2653.4 


•1 


3331.0 


2664.9 


• i 


3343.8 


2676.3 




3356.7 


2687.8 


'k' 


3369.5 


2699.3 


5 
• 8 


3382.4 


2710.8 


3 
•5 


3395.3 


2722.4 


7 
•8 


3408.2 


2733.9 


66. 


3421.2 


2745.5 


•i 


3434.1 


2757.1 


a 


3447.1 


2768.8 


.1 


3460.1 


2780.5 




3473.2 


2792.2 


•I 


3486.3 


2803.9 


.4 


3499.3 


2815.6 


7 
•8 


3512.5 


2827.4 


67. 


3525.6 


2839.2 


.i 


3538.8 


2851.0 


.i 


3552.0 


2862.8 


.i 


3565.2 


2874.7 


A 


3578.4 


2886.6 




3591.7 


2898.5 


.z 


3605.0 


2910.5 


.1 


3618.3 



Diameter 


Area. 


Diameter 


A 


68. 


3631.6 


75. 


44 


•i 


3645.0 




44 


• i 


3658.4 


, - . 


44^ 


•f 


3671.8 


•t 


44 




3685.2 


•t 


44' 


•1 


3698.7 




44< 


.§ 


3712.2 


.5 


45( 


7 
* 8 


3725.7 


• ■§■ 


455 


69. 


3739.2 


76. 


45: 


• a 


3752.8 


J 
•8 


45^ 


4 


3766.4 


•i 


45f 


3 

• 3 


3780.0 


3 
•8 


45^ 




3793.6 


.f 


45^ 


•1 


3807.3 


•I 


46J 


.£ 


3821.0 


A 


465 


•f 


3834.7 


.1 


464 


70. 


3848.4 


77. 


46£ 


,^- 


.3862.2 


4 


467 


. 2 


3875.9 


• i 


468 


.- 


3889.8 


•f 


470 


.i 


3903.6 




471 


,1 


3917.4 


• f 


473 




3931.3 


.4 


474 


1 


3945.2 


7 
• 8 


476 


71. 


3959.2 


78. 


477 


•1 


3973.1 


.1 


479 


a 


3987.1 


:i 


480 


3 


4001.1 




482 


"1 


4015.1 


i 


483 


5 


4029.2 


4 


485 


•1 


4043.2 


.i 


487 


7 
•8 


4067.3 


7 
.8 


488 


72. 


4071.5 


79. 


490 


•i 


4085.6 


.1 


491 


• i 


4099.8 


•t 


493 


•f 


4114.0 




494 




4128.2 


' ^ 


496 


•1 


4142.5 


!i 


497 


a 


4156.7 


A 


499 




4171.0 


7 
•8 


501 


73. 


4185.3 


80. 


502 


4 


4199.7 


.8 


504 


.i 


4214.1 


.i 


505 




4228.5 




507 


•i 


4242.9 


, Y 


508 


.1 


4257.3 




510 


.1 


4271.8 


. z 


512 


•1 


4286.3 


7 
.8 


513 


74. 


4300.8 


81. 


515 


•8 


4315.3 


.1 


516 




4329.9 


.i 


518 


.^ 


4344.5 


•i 


520 


.^ 


4359.1 


,1 


.521 


•¥ 


4373.8 


5 


523 


.i 


4388.4 


•1 


524 


•¥ 


4403.1 


•i' 


526 



94 



AREAS OF CIRCLES. 



Table— (Continued). 



D a'Tneter 


Area. 


Diameter. 


Area. | Diame 


ter. 


Area. 


Diameter. 


Area. 


82. 


5281.0 


87. 


5944.6 92. 




6647.6 


97. 


7389 




5297.1 


4 


5961.7 


I 


6665.7 


'i 


7408 


1 


5313.2 


'i 


5978.9 


i 


6683.8 


'4 


7427 


.*! 


5329.4 




5996.0 


i 


6701.9 


3 

•3 


7447 


• i 


5345.6 


.i 


6013.2 


k 


6720.0 


•! 


7466 




5331.8 


.1 


6030.4 


1 


6738.2 


h 


7485 


• 1 


5378.0 


•f 


6047.6 


1 


6756.4 


.| 


7504 




5394.3 


. ^ 


6064.8 


7 
8 


6776.4 


•f 


7523 


83. 


5410.6 


88." 


6082.1 93. 




6792.9 


98. 


7542 


.| 


5426.9 


.1 


6099.4 


r 


6811.1 


•8 


7562 


.5 


5443.2 


•i 


6116.7 


]'■ 


6829.4 


.? 


7581 




5459.6 


3 
• S 


6134.0 




6847.8 




7600 


*| 


5476.0 


• i 


6151.4 




6866.1 


• Y 


7620 


6 
. 8 


5492.4 


.1 


6168.8 


■• 


6884.5 




7639 


• 4 


5508.8 


•t 


6186.2 




6902.9 


'Z 


7658 


• 'k 


5525.3 


• i 


6203.6 


r 


6921.3 


• 8 


7678 


84. 


5541.7 


89. 


6221.1 94 




6939.7 


99. 


7697 


.1 


5558.2 


1 

• 8 


6238.6 


i 


6958.2 


• i 


7717 


.i 


5574.8 


•t 


6256.1 


I 


6976.7 


.i 


7736 




5591.3 


•i 


6273.6 


3 


6995.2 


.^ 


7756 


*!. 


5607.9 


.i 


6291.2 


^ 


7013.8 


.^ 


7775 


* & 


5624.5 


5 


6308.8 


1 


7032.3 


•t 


7795 


4 


5641.1 


•1 


6326.4 


I 


7050.9 


•4 


7814 


7 


5657.8 


• i 


6344.0 


a 


7069.5 


7 
•3 


7834 


. g 

85. 


5674.5 


w. 


6361.7 05 




7088. 2 


100. 


7853 


4 


5691.2 


,1 


6:579.4 




7106.9 






4 


5707.9 


.1 


6397.1 


i 


7125.5 






•■J 


5724.6 


•7T 


6114.8 


^ 


7144.3 






.^ 


5741.4 


4 ■ 


64.52.6 


k 


7163.0 






•8 


575S.2 


•1 


6450.4 


5 
8 


7181.8 






•1 


5775 


.1 


6468.2 


i 


7200.5 






7 
•8 


5791.9 


7 
•8 


6486.0 


7 
8 


7219.4 






86. 


5808.8 


91.' 


6503.8 96 




7238.2 






1 

•8 


5825.7 




6521.7 


1 


7257.1 






.? 


5842.6 




6539.6 


i 


7275.9 






3 


5859.5 


• i 


6557.6 


3 
•8 


7294.9 






• i 


5876.5 


■ .1 


6575.5 


.i 


7313.8 






'l 


5893.5 


• f 


6593.5 


.1 


7332.8 






3 


5910.5 


• 1 


6611.5 


.i 


7351.7 






7 
•8 


5927.6 


7 
•8 


6629.5 


7 
>8 


7370.7 







CIRCrMFEEENCES OF CIRCLES. 



95 



Circumferences of Circles, from 1 to 100. 



Dumeter 


Circumference 


Diameter. 


Circumference 


Diameter 


Circumference 


.1 Diameter. 


Circumfer'ce 


I 


.0490 


5. 


15.70 


12. 


37.69 


19. 


59.69 


.0981 


.1 


16.10 


4 


38.09 


4 


60.08 


■h 


.i 


16.49 


4 


38.48 


,i 


60.47 


fV 


.1963 


•8 


16.88 


.1 


38.87 




60.86 


1 


.3926 


•t 


17.27 


• i 


39.27 


,i- 


61.26 






. 3 


17.67 


•1 


39.66 


^5 


61.65 


■h 


.5890 


• i 


18.06 


1 


40.05 


'1 


62.04 


I 


.7854 


7 
• 3 


18.45 


•1 


40.44 


*| 


62.43 


5 


.9817 


6. 


18.84 


13. 


40.84 


20.' 


62.83 


18" 


'i 


19.24 


1 

• 8 


41.23 


4 


63.22 


f 


1.178 


.i 


19.63 




41.62 


'i 


63.61 


tV 


1.374 


• 8 


20.02 


•f 


42.01 


4 


64.01 






• i 


20.42 


•i 


42.41 


A 


64.40 


i 


1.570 


.1 


20.81 


•I 


42.80 


5 
• 8 


64.79 


9 


1.767 


•f 


21.20 


.4 


43.19 


.! 


65.18 


5 




• i 


21.57 


7 
• 8 


43.58 


4 


65.58 


■ff 


1.963 


7. 


21.99. 


14. 


43-. 98 


21.' 


65.97 


1 I 

TS" 


2.159 


1 
•3 


22.38 


.i 


44.37 


'i 


66.36 


3 


2.356 


•i 


22.77 


.i 


44.76 


•i 


6^.75 


T 


'-S 


23.16 


.1 


45.16 




67.15 


tI 


2.552 


'i 


23.56 


A 


45.55 


• 1 


67.54 


7 


2.748 


•f 


23.95 


.1 


45.94 


• 8 


67.93 


1 c 


2.945 




24.34 


.i 


46.33 


'i 


68.32 


tI 


7 
• S 


24.74 


.1 


46.73 


•1 


68.72 


1. 


3.141 


8. 


25.13 


15. 


47.12 


22.] 


69.11 






3.534 


.1 


25.52 


.1 


47.51 




69.50 






3.927 


•t 


25.91 


.i 


47.90 


^_ 


69.90 






4.319 




26.31 


.i 


48.30 


.- 


70.29 






4.712 


• i 


26.70 


A 


48.69 


• r 


70.68 






5.105 


.1 


27.09 


.8 


49.08 


•8 


71.07 






5.497 


.1 


27.48 


.1 


49.48 


.4 


71.47 






5.890 


7 
•8 


27.88 


7 
.8 


49.87 


• i 


71.86 


2 




6.283 


9. 


28.27 


16. 


50.26 


23.' 


72.25 






6.675 


•8 


28.66 


.i 


50.65 


.1 


72.64 






7.068 


•5 


29.05 


.i 


51.05 


.i 


73.04 






7.461 


.- 


29.45 


.f 


51.44 


.t 


73.43 






7.854 


.1 


29.84 




51.83 


.i 


73.82 






8.246 


•1 


30.23 


.1 


52.22 


.1 


74.21 






8.639 


•1 


30.63 


.| 


52.62 


.i 


74.61 






9.032 


• I 


31.02 


.1 


53.01 


'I 


75. 


3 




9.424 


10. 


31.41 


17. 


53.40 


24.] 


75.39 






9.817 


•1 


31.80 


.1 


53.79 




75.79 






10.21 


• 1 


32.20 


.i 


54.19 


*^ 


76.18 






10.60 


•^ 


32.59 


.1 


54.58 


•f 


76.57 






10.99 




32.98 


.•^ 


54.97 




76.96 






11.38 


•1 


33.37 


.1 


55.37 


• 1 


77.36 






11.78 


• ^ 


33.77 


.1 


55.76 


.? 


77.75 






12.17 


•1 


34.16 


7 
.8 


56.16 


• ? 


78.14 


Jc 




12.56 


11. 


34.55 


18. 


56.54 


25. 


78.54 






12.95 


'i 


34.95 


.1 


56.94 


4 


78.93 






13.35 


•i 


35.34 




57.33 


'i 


79.32 






13.74 


•5 


35.73 


.i 


57.72 


.f 


79.71 






14.13 


•^ 


36.12 


.1 


58.11 




80.10 






14.52 


•1 


36.52 


.1 


.58.51 


•1 


80.50 






14.92 


4 


36.91 
37.30 


.i 


58.90 


^ 


80.89 




i 


15.31 


7 


7 
•5 


59.29 


4 


81.28 



96 



CIRCUMFERENCES OF CIRCLES. 



Table— (Continued). 



Diameter Circumference. Diameter. ! Circumference 



81.68 

82.07 
82.46 
82.85 
83.25 
83.64 
84.03 
84.43 
84.82 
85.21 
85.60 
86. 
86.39 
86.78 
87.17 
87.57 
87.96 
88.35 
88.75 
89.14 
89.53 
89.92 
90.32 
90.71 
91.10 
91.49 
91.89 
92.28 
92.67 
93.06 
93.46 
93.85 
94.24 
94.64 
95.03 
95.42 
95.81 
96.21 
96.60 
96.99 
97.38 
97.78 
98.17 
98.56 
98.96 
99.35 
99.74 
100.1 
100.5 
100.9 
101.3 
101.7 
102.1 
102.4 
102.8 
103.2 



33 



34 



35 



36 



37 



38 



39 



103.6 

104. 

104.4 

104.8 

105.2 

105.6 

106. 

106.4 

106.8 

107.2 

107.5 

107.9 

108.3 

108.7 

109.1 

109.5 

109.9 

110.3 

110.7 

111.1 

111.5 

111.9 

112.3 

112.7 

113. 

113.4 

113.8 

114.2 

114.6 

115. 

115.4 

115.8 

116.2 

116.6 

117. 

117.4 

117.8 

118.2 

118.6 

118.9 

119.3 

119.7 

120.1 

120.5 

120.9 

121.3 

121.7 

122. 1 

122.5 

122.9 

123.3 

123.7 

124. 

124.4 

124.8 

125.2 



Diameter. 


Circumference. Diame 


ter. 


Circumfer'ce 


40. 


125.6 47. 




147.6 




-I 


126. 


- 


148. 




i 


126.4 


5 


148.4 






126.8 


j; 


148.8 




1 


127.2 


^ 


149.2 




1 


127.6 


1 


149.6 




I 


128. 


f 


150. 




i 


128.4 


1 


150.4 


41 


128.8 48 




150.7 




1 


129.1 


- 


151.1 




1 


129.5 


_ 


151.5 




I 


129.9 


i 


151.9 




8 


130.3 


I 


152.3 




Jl 


130.7 


1 


152.7 




S 
.1 


131.1 


1 


153.1 




2 


131.5 


i 


153.5 


42 


/ 


131.9 49 




153.9 




'i 


132.3 


... 


154.3 






132.7 


,]. 


154.7 




f 


133.1 


•8 


155.1 




I 


133.5 


,| 


155.5 




1 


133.9 


5 


1.55.9 




i 


134.3 


.1 


166.2 




■^ 


134.6 


7 
• 8 


156.6 


43 




135. 50 




157. 




'i 


135.4 


•r 


157.4 






135.8 


. :: 


157.8 




2. 


136.2 


/A 


158.2 




I 


136.6 


.- ■ 


158.6 




b 


137. 


,'.. 


1.59. 




I 


137.4 


Jy 


159.4 




I 


137.8 


• 1 ■ 


159.8 


44 


8 


138.2 51 




160.2 




I 


138.6 


*i 


160.6 






139. 


• i 


161. 




3 


139.4 




161.3 




^- 


139.8 


.^ 


161.7 




1 


140.1 


1 


162.1 




140.5 




162.5 




7 


140.9 


i 


162.9 


45 


g 


141.3 52 




163.3 




1 


141.7 


'i 


163.7 




i 


142.1 


i 


164.1 






142.5 


8 


164.5 




1 


142.9 


i 


164.9 




1 


143.3 


5 


165.3 




1 


143.7 


1 


165.7 






144.1 


7 

5 


166.1 


46 




144.5 53 




166.5 




i 


144.9 


'l 


166.8 




i 


145.2 


4 


167.2 




3 
8 


145.6 


8 


167.6 




.V 


146. 


1 


168. 






146.4 




168.4 




I 


146.8 


3 


168.8 




i 


147.2 


i 


169.2 



CIRCUMFERENCES OF CIRCLES. 



97 



Table — (Continued). 



Diamete 


r Circumference 


Diameter 


1 Circumference 


Diameter. 


Circumference 


. Diameter 


[Circu 


54. 


169.6 


61. 


191.6 


68. 


213.6 


75. 


23 


.| 


170. 


.1 


192. 


•1 


214. 


•8 


23 


•i 


170.4 


•i 


192.4 


'i 


214.4 


• i 


23 


3 

•8 


170.8 


•f 


192.8 


•8 


214.8 


.f 


23 




171.2 




193.2 


•t 


215.1 


•t 


23 


•1 


171.6 


•1 


193.6 




215.5 


*t 


23 


.1 


172. 


•1 


193.9 


•1 


215.9 


.i 


23 


.J 


172.. 3 


•i 


194.3 


.| 


216.3 


7 


23 


55. 


172.7 


62. 


194.7 


69. 


216.7 


76? 


23 


'i 


173.1 


'i 


195.1 


•1 


217.1 


•f 


23 


.1 


173.5 


.1 


195.5 


•1 


217.5 




23 


3 
•^ 


173.9 




195.9 


•i 


217.9 


• i 


23 


• i 


174.3 


•1 


196.3 


.1. 


218.3 




24 


5 

•8 


174.7 


5 

• 8 


196.7 


4 


218.7 


•f 


24 




175.1 


•J 


197.1 


•f 


219.1 


A 


24 


1 


/175.5 


7 
• 3 


197.5 


• i 


219.5 


.1 


24 


56. 


175.9 


63. 


197.9 


70. 


219.9 


77. 


24 


'i 


176.3 


1 

•8 


198.3 


1 

•8 


220.3 


.1 


24 


a 


176.7 




198.7 


1 
• 4 


220.6 


1 


24* 


.f 


177.1 


•f 


199. 


• 8 


221. 


3 

•8 


24: 


•i 


177.5 


•2" 


199.4 


•f 


221.4 




24: 


^i 


177.8 


•f 


199.8 




221.8 


• ¥ 


24: 


•1 


178.2 


.| 


200.2 


• 1 


222.2 


•f 


24^ 


,1 


178.6 


7 
•8 


200.6 


,1 


222.6 


•3 


24^ 


57. 


179. 


64. 


201. 


71.' 


223. 


78. 


24^ 


,- 


179.4 


•i 


201.4 


•1 


223.4 


.1 


24f 


•4 


179.8 


•5 


201.8 


.i 


223.8 


.1 


24f 


.- 


180.2 


•t 


202.2 


• w 


224.2 


• I 


24( 


•i 


180.6 


4 


202.6 


•t 


224.6 


A 


24( 


.- 


181. 


• s 


203. 




225. 


.1 


24i 


.1 


181.4 


.1 


203.4 


• 1 


225.4 


.1 


24'3 


•i 


181.8 


7 
•8 


203.8 


7 
•8 


225.8 


7 
.¥ 


247 


58. 


182.2 


65. 


204.2 


72. 


226.1 


79. 


248 


.1 


182.6 


•1 


204.5 


.1 


226.5 


• 1 


248 


•i 


182.9 


•t 


204.9 


.i 


226.9 


• ? 


248 


;i 


183.3 




205.3 




227.3 


.i 


24S 


. \ 


183.7 


'i 


205.7 


."1 


227.7 


.* 


24S 


5 
•8 


184.1 


5 


206.1 


5 

• 3 


228.1 


1 


25C 


.f 


184.5 


•5 


206.5 


.1 


228.5 


*| 


25G 


7 


184.9 


7 
•8 


206.9 




228.9 


* 7 
.5 


25G 


39. 


185.3 


66. 


207.3 


73!" 


229.3 


80. 


251 


•i 


185.7 


•1 


207.7 


,1 


229.7 


.i 


251 


•i 


186.1 


•i 


208.1 


•| 


230.1 


.i 


252 


.•2 


186.5 


.i 


208.5 


••^ 


230.5 


.8 


252 


• i 


186.9 


•t 


208.9 


4 


230.9 


.i 


252 


• 1 


187.3 




209.3 


• 8 


231.3 


.3 


253 


.f 


187.7 


.^ 


209.7 




231.6 


. z 


253 


4 


188.1 


.1 


210. 


7 
•8 


2.32. 


.1 


254 


50. 


188.4 


67. 


210.4 


74. 


232.4 


81. 


254 


■ i 


188.8 


.1 


210.8 


• i 


232.8 


.8 


254 


1 


189.2 


.i 


211.2 


.| 


233.2 


^; ■ 


255 


[a 


189.6 


• 1 


211.6 




233.6 


• i 


255 


•i 


190. 


.2- 


212. 


,^ 


234. 


•I 


256 


•8 


190.4 


.8 


212.4 


5 


234.4 


ft 


256 


n 

•^ 


190.8 


.1 


212.8 


*g 


234.8 


.1 


256 


7 


191.2 


7 

•5 


213.2 


'■1 


235.2 


4 


257 



98 



CIRCUMFERENCES OF CIRCLES. 



Table— (Continued). 



Diameter 


Circumference. 


Diameter. 


Circumference. 


Diameter. 


Circumference. 


Diameter." 


Circumfer'ca 


82. 


257.6 


87. 


273.3 


92. 


289. 


97. 


304.7 




258. 


'i 


273.7 


.1 


289.4 


■ i 


305.1 


258.3 


,1 


274.1 


4 


289.8 


I 


305.5 


3 


258.7 


2 


274.4 


• i 


290.2 


*.| 


305.9 


• i 


259.1 


.1 


274.8 


4 


290.5 


'k 


306.3 


i 


259.5 


•1 


275.2 


ft 

• 8 


290.9 


•1 


306.6 


259.9 


.1 


275.6 


.1 


291.3 


.1 


307. 


7 


260.3 


7 
• S 


276. 


7 
•8 


291.7 


7 
•8 


307.4 


83. 


260.7 


88. 


276.4 


93. 


292.1 


98. 


307.8 


4 


261.1 


.1 


276.8 


.1 


292.5 


'i 


308.2 




261.5 


4 


277.2 


.i 


292.9 


• ? 


308.6 


• 1 


261.9 


• s 


277.6 




293.3 


• i 


309.0 


i| 


262.3 




278. 


• i 


293.7 


•t 


309.4 


.1 


262.7 


•1 


278.4 


• 1 


294.1 


.1 


309.8 


A 


263.1 


•f 


278.8 


.? 


294.5 


.^ 


310.2 


7 


263.5 


• t 


279.2 


7 
•8 


294.9 


7 
•8 


310.6 


84. 


263.8 


89. 


279.6 


94. 


295.3 


99. 


311.0 




264.2 


•8 


279.9 


.1 


295.7 


• i 


311.4 


_ 


264.6 




280.3 


1 


296. 


^1 


311.8 


•f 


265. 


•¥ 


^80.7 


,f 


296.4 


• i 


312.1 




265.4 


,i. 


281.1 


• i 


296.8 


.^ 


312.5 


• 1 


265.8 


.1 


281.5 


•1 


297.2 


•■J 


312.9 


• 1 


266.2 


.1 


281.9 


.? 


297.6 


^ J 


313.3 


•1 


266.6 


7 
•8 


282.3 


•i- 


298. 


• i 


313.7 


8.5*.' 


267. 


90. 


282.7 


95. 


298.4 


100. 


314.1 




267.4 


'i 


283.1 


4 


298.8 






• ? 


267.8 


• i 


283.5 


• i 


299.2 






• i 


268.2 


•t 


283-9 


3 
•8 


299.6 






•t 


269.6 


•i 


284.3 


•i 


300. 








268.9 


.1 


284.7 


.1 


300.4 






• 1 


269.3 


4 


285.1 


•t 


«00.8 






7 
. 8 


269.7 


7 
•8 


285.4 


• i 


301.2 






86. 


270.1 


91. 


285.8 


96. 


301.5 






.1 


270.5 


•1 


286.2 


•8 


301.9 








270.9 


• i 


286.6 


•? 


302.3 






'! 


271.3 


3 


287. 


•8 


302.7 






• r 


271.7 


.^ 


287.4 


.-- 


303.1 






• 1 


272.1 


•t 


287.8 




303.5 






.i 


272.5 


•I 


288.2 


. J: 


303.9 






4 


272.9 


■l 


288.6 


.| 


304.3 







SQUARES, CUBES, AND ROOTS. 



99 



Table of Squares, Cubes, and Square and Cube Roots, of all lumbers 
from 1 to 1000. 

Number. Square. 



1 


1 


4 


8 


9 


27 


16 


64 


25 


125 


36 


216 


49 


343 


64 


512 


81 


729 


100 


1000 


121 


1331 


144 


1728 


169 


2197 


196 


2744 


225 


3375 


256 


4096 


289 


4913 


324 


. 5832 


361 


6859 


400 


8000 


441 


9261 


484 


10648 


529 


12167 


576 


13824 


625 


15625 


676 


17576 


729 


19683 


784 


21952 


841 


24389 


900 


27000 


961 


29791 


1024 


32768 


1089 


35937 


1156 


39304 


1225 


42875 


1296 


46656 


1369 


50653 


1444 


54872 


1521 


59319 


1600 


64000 


1681 


68921 


1764 


74088 


1849 


79507 


1936 


85184 


2025 


91J25 


2116 


97336 


2209 


103823 


2304 


110592 


2401 


117649 


2500 


125000 


2601 


132651 


2704 


140608 


2809 


148877 


2916 


157464 


3025 


166375 



Square Root, 



1. 

1.414213 

1.732050 

2. 

2.236068 

2.449489 

2.645751 

2.828427 

3. 

3.162277 

3.316624 

3.464101 

3.605551 

3.741657 

3.872983 

4. 

4.123105 

4.242640 

4.358898 

4.472136 

4.582575 

4.690415 

4.795831 

4.898979 

5. 

5.099019 

5.196152 

5.291502 

5.385164 

5.477225 

5.567764 

5.656854 

5 . 744562 

5.830951 

5.916079 

6. 

6.082762 

6.164414 

6.244998 

6.324555 

6.403124 

6.480740 

6.557438 

6.633249 

6.708203 

6.782330 

6.855654 

6.928203 

7. 

7.071067 

7.141428 

7.211102 

7.280109 

7.348469 

7.416198 



Cube Root. 



1. 

1.259921 

1.442250 

1.587401 

1.709976 

1.817121 

1.912933 

2. 

2.080084 

2.154435 

2.223980 

2.289428 

2.351335 

2.410142 

2.466212 

2.519842 

2.571282 

2.620741 

2.668402 

2.714418 

2.758923 

2.802039 

2.843867 

2.884499 

2.924018 

2.982496 

3. 

3.036589 

3.072317 

3.107232 

3.141381 

3.174802 

3.207534 

3.239612 

3.271066 

3.301927 

3.332222 

3.361975 

3.391211 

3.419952 

3.448217 

3.476027 

3.503398 

3.530348 

3.556893 

3.583048 

3.608826 

3.634241 

3.659306 

3.684031 

3.708430 

3.732511 

3.756286 

3.779763 

3.802953 



100 



SQUARES, CUBES, AND ROOTS. 



Table — (Continued). 



Number. 


Square. 


Cube. i 


Square Root. 


56 


3136 


175616 


7.483314 


57 


3249 


185193 


7.549834 


58 


3364 


195112 


7.615773 


59 


3481 


205379 


7.681145 


60 


3600 


216000 


7.745966 


61 


3721 


226981 


7.810249 


62 


3844 


238328 


7.874007 


63 


3989 


250047 


7.937253 


64 


4096 


262144 


8. 


65 


4225 


274626 


8.062257 


66 


4356 


287496 


8.124038 


67 


4489 


300763 


8.185352 


68 


4624 


314432 


8.246211 


69 


4761 


328509 


8.306623 


70 


4900 


343000 


8.366600 


71 


5041 


357911 


8.426149 


72 


5184 


373248 


8.485281 


73 


5329 


389017 


8.544003 


74 


5476 


405224 


8.602325 


75 


5625 


421875 


8.660254 


76 


5776 


43S976 


8.717797 


77 


5929 


456533 


8.774964 


78 


6084 


474552 


8.831760 


79 


6241 


493039 


8.888194 


80 


6400 


512000 


8.944271 


81 


6561 


531441 


9. 


82 


6724 


551368 


9.055385 


83 


6889 


571787 


9.110433 


84 


7056 


592704 


9.165151 


85 


7225 


614125 


9.219544 


86 


7396 


636056 


9.273618 


87 


7569 


658503 


9.327379 


88 


7744 


6S1472 


9.380831 


89 


7921 


704969 


9.433981 


90 


8100 


729000 


9.486833 


91 


8281 


753571 


9.539392 


92 


8464 


778688 


9.591663 


93 


8649 


804357 


9.643650 


94 


8836 


830584 


9.695359 


95 


9025 


857375 


9.746794 


96 


9216 


.884736 


9.797959 


97 


9409 


912673 


9.848857 


98 


9604 


941192 


9.899494 


99 


9801 


970299 


9.949874 


100 


lOOOO 


1000000 


10. 


101 


10201 


1030301 


10.049875 


102 


10404 


1061208 


10.099504 


103 


10609 


1092727 


10.148891 


104 


10816 


1124864 


10.198039 


105 


11025 


1157625 


10.246950 


106 


11236 


1191016 


10-295630 


107 


11449 


1225043 


10.344080 


108 


11664 


1259712 


10.392304 


109 


11881 


1295029 


10.440306 


110 


12100 


1331000 


10-488088 


111 


12321 


1367631 


10.535653 



SQUARES, CUBES, AND ROOTS. 



101 



Table — (Continued). 



p Namber. 


Sqnare. 


1 Cube. 


Square Root. 


Cube Root. 


112 


12544 


1404928 


10.583005 


4.820284 


113 


12769 


1442897 


10.630145 


4.834588 


114 


12996 


1481544 


10.677078 


4.848808 


'115 


13225 


1520375 


10.723805 


4.862944 


116 


13456 


1560896 


10.770329 


4.876999 


117 


13689 


1601613 


10.816653 


4.890973 


118 


13924 


1643032 


10.862780 


4.904868 


119 


14161 


1685159 


10.908712 


4.918685 


120 


14400 


1728000 


10.954451 


4.932424 


121 


14641 


1771561 


11. 


4.946088 


122 


14884 


1815848 


11.045361 


4.959675 


123 


15129 


1860867 


11.090536 


4.973190 


124 


15376 


1906624 


11.135528 


4.986631 


125 


15625 


1953125 


11.180339 


5. 


126 


15876 


2000376 


11.224972 


5.013298 


127 


16129 


2048383 


11.269427 


5.026526 


128 


16384 


2097152 


11.313708 


5.039684 


129 


16641 


2146689 


11.357816 


5.052774 


130 


16900 


2197000 


11.401754 


5.065797 


131 


17161 


2248091 


11.445523 


5.078753 


132 


17424 


2299968 


11.489125 


5.091643 


133 


17689 


2352637 


11.532562 


5.104469 


134 


17956 


2406104 


11.575836 


5.117230 


135 


18225 


2460373 


11.618950 


5.129928 


136 


18496 


2515456 


11.661903 


5.142563 


137 


18769 


2571353 


11.704699 


5.155137 


138 


19044 


2628072 


11.747344 


5.167649 


139 


19321 


2685619 


11.789826 


5.180101 


140 


19600 


2744000 


11^832159 


5.192494 


141 


19881 


2803221 


11.874342 


5.204828 


142 


20164 


2863288 


11.916375 


5.217103 


143 


20449 


2924207 


11.958260 


5.229321 


144 


20736 


2985984 


12. 


5.241482 


145 


21025 


3048625 


12.041594 


5.253588 


146 


21316 


3112136 


12.083046 


5.265637 


147 


21609 


3176523 


12.124355 


5.277632 


148 


21904 


3241792 


12.165525 


5.289572 


149 


22201 


3307949 


12.206555 


5.301459 


150 


22500 


3375000 


12.247448 


5.313293 


151 


22801 


3442951 


12.288205 


5.325074 


152 


23104 


3511808 


12.328828 


5.336803 


153 


23409 


3581577 


12.369316 


5.348481 


154 


23716 


3652264 


12.409673 


5.360108 


155 


24025 


3723875 


12.449899 


5.371685 


156 


24336 


3796416 


12.489996 


5.3S3231 


157 


24649 


3869893 


12.529964 


5.394690 


158 


24964 


3944312 


12-569805 


5.406120 


159 


25281 


4019679 


12.609520 


5.417501 


160 


25600 


4090000 


12.649110 


5.428835 


161 


25921 


4173281 


12.688577 


5.440122 


162 


26244 


4251528 


12-727922 


5.451362 


163 


26569 


4330747 


12.767145 


5.462556 


164 


26896 . 


4410944 


12.806248 


5.473703 


165 


27225 


4492125 


12-845232 


5.484806 


166 


27556 


4574296 


12.884098 


5.495865 


167 


27889 


4657463 

12 


12-922848 


5.506879 



102 



SQUARES, CUBES, AND ROOTS. 



Table— (Continued). 



Square. 



28224 

28561 

28900 

29241 

29584 

29929 

30276 

30625 

30976 

31329 

31684 

32041 

32400 

32761 

33124 

33489 

33856 

34225 

34596 

34969 

35344 

35721 

36100 

36481 

36864 

37249 

37636 

38025 

38416 

38809 

39204 

39601 

40000 

40401 

40804 

41209 

41616 

42025 

42436 

42849 

43264 

43681 

44100 

44521 

44944 

45369 

45796 

46225 

46656 

47089 

47524 

47961 

48400 

48841 

49284 

49729 



Cube. 


Square Root. 


Cube Root. 


4741632 


12.961481 


5.517848 


4826809 
4913000 


13. 
13.038404 


5.528775 
5.539658 


5000211 


13.076696 


5.550499 


5088448 


13.114877 


5.561298 


5177717 


13.152946 


5.572054 


5268024 


13.190906 


5.582770 


5359375 


13.228756 


5.593445 


5451776 


13.266499 


5.604079 


5545233 


13.304134 


5.614673 


5639752 


13.341664 


5.625226 


5735339 


13.379088 


5.635741 


5832000 


13.416407 


5.646216 


5929741 


13.453624 


5.656652 


6028568 


13.490737 


5.667051 


6128487 


13.527749 


5.677411 


6229504 


13.564660 


5.687734 


6331625 


13.601470 


5.698019 


6434856 


13.638181 


5.708267 


6539203 


13.674794 


5.718479 


6644672 


13.711309 


5.728654 


6751269 


13.747727 


5.738794 


6859000 


13.784048 


5.748897 


6967871 


13.820275 


5.758965 


7077888 


13.856406 


5.768998 


7189057 


13.892444 


5.778996 


7301384 


13.928388 


5.788960 


7414875 


13.964240 


5.798890 


7529536 


14. 


5.808786 


7645373 


14.035668 


5.818648 


7762392 


14.071247 


5.828476 


7880599 


14.106736 


5.838272 


8000000 


14.142135 


5.848035 


8120601 


14.177446 


5.857765 


8242408 


14.212670 


5.867464 


8365427 


14.247806 


5.877130 


8489664 


14.282856 


5.886765 


8615125 


14.317821 


5.896368 


8741816 


14.352700 


5.905941 


8869743 


14.387494 


5.915481 


8998912 


14.422205 


5.924991 


9123329 


14.456832 


5.934473 


9261000 


14.491376 


5.943911 


9393931 


14.525839 


5.953341 


9528128 


14.560219 


5.962731 


9663597 


14.594519 


5.972091 


9800344 


14.628738 


5.981426 


9938375 


14.662878 


5.990727 


10077696 


14.696938 


6. 


10218313 


14.730919 


6.009244 


10360232 


14.764823 


6.018463 


10503459 


14.798648 


6.027650 


10648000 


14.832397 


6.036811 


10793861 


14.866068 


6,045943 


10941048 


14.899664 


6.055048 


11089567 


14.933184 


6.064126 



SQUARES, CUBES, AND ROOTS. 



103 



Table — (Continued). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


224 


50176 


11239424 


14.966629 


6.073177 


225 


50625 


11390625 


15. 


6.082201 


226 


51076 


11543176 


15.033296 


6.091199 


227 


51529 


11697083 


15.066519 


6.100170 


228 


51984 


11852352 


15.099668 


6.109115 


229 


52441 


12008989 


15.132746 


6.118032 


230 


52900 


12167000 


15.165750 


6.126925 


231 


53361 ^ 


12326391 


15.198684 


6.135792 


232 


53824 


12487168 


15.231546 


6.114634 


2.33 


54289 


12649337 


15.264337 


6.153449 


234 


54756 


12812904 


15.297058 


6.162239 


235 


55225 


12977875 


15.329709 


6.171005 


236 


55696 


13144256 


15.362291 


6.179747 


237 


56169 


13312053 


15.394804 


6.188463 


238 


56644 


13481272 


15.427248 


6.197154 


239 


57121 


13651919 


15.459624 


6.205821 


240 


57600 


13824000 


15.491933 


6.214464 


241 


58081 


13997521 


15.524174 


6.223083 


242 


58564 


14172488 


15.556349 


6.231678 


243 


59049 


14348907 


15.588457 


6.240251 


244 


59536 


14526784 


15.620499 


6.248800 


245 


60025 


14706125 


15.652475 


6.257324 


246 


60516 


14886936 


15.684387 


6.26.5826 


247 


61009 


15069223 


15.716233 


6.274304 


248 


61504 


15252992 


15.748015 


6.282760 


249 


62001 


15438249 


15.779733 


6.291194 


250 


62500 


15625000 


15.811388 


6.299604 


251 


63001 


15813251 


15.842979 


6.307992 


252 


63504 


16003008 


15.874507 


6.316359 


253 


64009 


16194277 


15.905973 


6.324704 


254 


64516 


16387064 


15.937377 


6.333025 


255 


65025 


16581375 


15.968719 


6.341325 


256 


65536 


16777216 


16. 


6.349602 


257 


66049 


16974593 


16.031219 


6.357859 


258 


66564 


17173512 


16.062378 


6.366095 


259 


67081 


17373979 


16.093476 


6.374310 


260 


67600 


17576000 


16.124515 


6.382504 


261 


68121 


17779581 


16.155494 


6.390676 


262 


68644 


17984728 


16.186414 


6.398827 


263 


69169 


18191447 


16.217274 


6.406958 


264 


69696 


18399744 


16.248076 


6.41.5068 


265 


70225 


18609625 


16.278820 


6.423157 


266 


70756 


18821096 


16.309506 


6.431226 


267 


71289 


19034163 


16.340134 


6.439275 


268 


71824 


19248832 


16.370705 


6.447305 


269 


72361 


19465109 


16.401219 


6.455314 


270 


72900 


19683000 


16.431676 


6.463304 


271 


73441 


19902511 


16.462077 


6.471274 


272 


73984 


20123648 


16.492422 


6.479224 


273 


74529 


20346417 


16.522711 


6.487153 


274 


75076 


20570824 


16.552945 


6.495064 


275 


75625 


20796875 


16.583124 


6.502956 


276 


76176 


21024576 


16.613247 


6.510829 


277 


76729 


21253933 


16.643317 


6.518684 


278 


77284 


21484952 


16.673332 


6.526519 


279 


77841 


21717639 


16.703293 


6.634335 



104 



SQUARES, CUBES, AND ROOTS. 



Table — (Continued). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


280 


78400 


21952000 


16.733200 


6.542132 


281 


78961 


22188041 


16.763054 


6.549911 


282 


79524 


22425768 


16.792855 


6.557672 


283 


80089 


22665187 


16.822603 


6.565415 


284 


80656 


22906304 


16.852299 


6.573139 


285 


81225 


23149125 


16.881943 


6.580844 


286 


81796 


23393656 


16.911534 


6.588531 


287 


82369 


23639903 


16.941074 


6.596202 


288 


82944 


23887872 


16.970562 


6.603854 


289 


83521 


24137569 


17. 


6.611488 


290 


84100 


24389000 


17.029386 


6.61910S 


291 


84681 


24642171 


17.058722 


6.626705 


292 


85264 


24897088 


17.088007 


6.634287 


293 


85849 


25153757 


17.117242 


6.641851 


294 


86436 


25412184 


17.146428 


6.649399 


295 


87025 


25672375 


17.175564 


6.656930 


296 


87616 


25934336 


17.204650 


6.664443 


297 


88209 


26198073 


17.233687 


6.671940 


298 


88804 


26463592 


17.262676 


6.679419 


299 


89401 


26730899 


17.291616 


6.686882 


300 


90000 


27000000 


17.320508 


6.694328 


301 


90601 


27270901 


17.349351 


6.701758 


302 


91204 


27543608 


17.378147 


6.70917^ 


303 


91809 


27818127 


17.406895 


6.716569 


304 


92416 


28094464 


17.435595 


6.723950 


305 


93025 


28372625 


17.464249 


6.73131(> 


306 


93636 


28652616 


17.492855 


6.738665 


307 


94249 


28934443 


17.521415 


6.745997 


308 


94864 


29218112 


17.549928 


6.753313 


309 


95481 


29503629 


17.578395 


6.760614 


310 


96100 


29791000 


17.608816 


6.767899 


311 


96721 


30080231 


17.635192 


6.775168 


312 


97344 


30371328 


17.663521 


6.782422 


313 


97969 


30664297 


17.691806 


6.7S9661 


314 


98596 


30959144 


17.720045 


6.796884 


315 


99225 


31255875 


17.748239 


6.804091 


316 


99856 


31554496 


17.776388 


6.S112S4 


317 


100489 


31855013 


17.804493 


6.818461 


318 


101124 


32157432 


17.832554 


6.825624 


319 


101761 


32461759 


17.86057.1 


6.832771 


320 


102400 


32768000 


17.888543 


6.839903 


321 


103041 


33076161 


17.916472 


6.847021 


322 


103684 


33386248 


17.944358 


6.854124 


323 


104329 


33698267 


17.972200 


6.861211 


324 


104976 


34012224 


18. 


6.868284 


325 


105625 


34328125 


18.027756 


6.87.5343 


326 


106276 


34645976 


18.055470 


6.882388 


327 


106929 


34965783 


18.083141 


6.889419 


328 


107584 


35287552 


18.110770 


6.896435 


329 


108241 


35611289 


18.138357 


6.903436 


330 


108900 


35937000 


18.165902 


6.910423 


331 


109561 


36264691 


18.193405 


6.917396 


332 


110224 


36594368 


18.220867 


6.924355 


333 


110889 


36026037 


18.248287 


6.931300 


334 


111556 


37259704 


18.275666 


6.938232 


335 


U2225 


87595375 


18.303005 


6.945149 



SQUARES, CUBES, AND ROOTS. 



105 



Table — (Continued). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


336 


112896 


37933056 


18.330302 


6.952053 ' 


337 


113569 


38272753 


18.357559 


6.958943 


338 


114244 


38614472 


18.384776 


6.965819 


339 


114921 


3S958219 


18.411952 


6.972682 


340 


115600 


39304000 


18.439088 


6.979532 


341 


116281 


39651821 


18.466185 


6.986369 


342 


116964 


40001688 


18.493242 


6.993491 


343 


117649 


40353607 


18.520259 


7. 


344 


118336 


40707584 


18.547237 


7.006796 


345 


119025 


41063625 


18.574175 


7.013579 


346 


119716 


41421736 


18.601075 


7.020349 


347 


120409 


41781923 


18.627936 


7.027106 


348 


^ 121104 


42144192 


18.654758 


7.033850 


349 


121801 


42508549 


18.681541 


7.040581 


350 


122500 


42875000 


18.708288 


7.047208 


351 


123201 


43243551 


18.7.34994 


7.054003 


352 


123904 


43614208 


18.761663 


7.060696 


353 


124609 


43986977 


18.788294 


7.067376 


354 


125316 


44361864 


18.814887 


7.074043 


355 


126025 


44738875 


18.841443 


7.080698 


356 


126736 


45118016 


18.867962 


7.087341 


357 


127449 


45499293 


18.894443 


7.093970 


358 


128164 


45882712 


18.920887 


7.100588 


350 


12SS81 


46268279 


18.947295 


7.107193 


360 


129600 


46656000 


18.973666 


7.113786 


361 


130321 


47045881 


19. 


7.120367 


362 


131044 


47437928 


19.026297 


7.126935 


363 


131769 


47832147 


19.052558 


7.133492 


364 


132496 


48228544 


19.078784 


7.140037 


365 


133225 


48627125 


19.104973 


• 7.146569 


366 


133956 


49027896 


19.131126 


7.153090 


367 


134689 


49430863 


19.157244 


7.159599 


368 


135424 


49836032 


19.183326 


7.166095 


369 


136161 


50243409 


19.209372 


7.172580 


370 


136900 


50653000 


19.235384 


7.179054 


371 


137641 


51064811 


19.261360 


7.185516 


372 


138384 


51478848 


19.287301 


7.191966 


373 


139129 


51895117 


19.313207 


7.198405 


374 


139876 


52313624 


19.339079 


7.204832 


375 


140625 • 


52734375 


19.364916 


7.211247 


376 


141376 


53157376 


19.390719 


7.217652 


377 


142129 


535S2633 


19.416487 


7.224045 


378 


142884 


54010152 


19.442222 


7.230427 


379 


143641 


54439939 


19.467922 


7.236797 


380 


144400 


54872000 


19.493588 


7.243156 


381 


145161 


55306341 


19.519221 


7.249504 


382 


145924 


55742968 


19.544820 


7.255841 


383 


146689 


56181887 


19.570385 


7.262167 


384 


147456 


56623104 


19.595917 


7.268482 


385 


148225 • 


57066625 


19.621416 


7.274786 


386 


148996 


57512456 


19.646882 


7.281079 


387 


149769 


57960603 


19.672315 


7.287362 


388 


150544 


58411072 


19.697715 


7.293633 


389 


151321 


58863869 


19.723082 


7.299893 


390 


152100 


59319000 


19.748417 


7.306143 


391 


152881 


59776471 


19.773719 


7.312383 



106 



SQUARES, CUBES, AND ROOTS. 





Table — (Continued). 




Square, 


Cube. 


Square Root. 


Cube Root. 


153664 


60236288 


19.798989 


7.318611 


154449 


60698457 


19.824227 


7.324829 


155236 


61162984 


19.849432 


7.331037 


156025 


61629S75 


19.874606 


7.337234 


156816 


62099136 


19.899748 


7.343420 


157609 


62570773 


19.924858 


7.349596 


158404 


63044792 


19.949937 


7.355762 


159201 


63521199 


19.974984 


7.361917 


160000 


64000000 


20. 


7.368063 


160801 


64481201 


20.024984 


7.374198 


161604 


64964808 


20.049937 


7.380322 


162409 


65450827 


20.074859 


7.386437 


163216 


65939264 


20.099751 


7.392542 


164025 


66430125 


20.124611 


7.398636 


164836 


66923416 


20.149441 


7.404720 


165649 


67419143 


20.174241 


7.410794 


166464 


67911312 


20.199009 


7.416859 


167281 


68417929 


20.223748 


7.422914 


168100 


68921000 


20.248456 


7.428958 


168921 


69426531 


20.273134 


7.434993 


] 69744 


69934528 


20.297783 


7.441018 


170569 


70444997 


20.322401 


7.447033 


171396 


70957944 


20.346989 


7.453039 


172225 


71473375 


20.371548 


7.459036 


173056 


71991296 


20.396078 


7.465022 


173889 


72511713 


20.420577 


7.470999 


174724 


73034632 


20.445048 


7.476966 


175561 


73560059 


20.469489 


7.482924 


176400 


74088000 


20.493901 


7.488872 


177241 


74618461 


20.518284 


7.494810 


178084 


75151448 


20.542638 


7.500740 


178929 


75686967 


20.566963 


7.506660 


179776 


76225024 


20.591260 


7.512571 


180625 


76765625 


20.615528 


7.5184-73 


181476 


77308776 


20.639767 


7.524365 


182329 


77854483 


20.663978 


7.530248 


183184 


78402752 


20.688160 


7.536121 


184041 


78953589 


20.712315 


7.541986 


184900 


79507000 


20.736441 


7.547841 


185761 


80062991 


20.760539 


7.553688 


186624 


80621568 


20.784609 


7.559525 


187489 


81182737 


20.808652 


7.565353 


188356 


81746504 


20.832666 


7.57117.a 


189225 


82312875 


20.856653 


7.576984 


190096 


82881856 


20.880613 


7.582786 


190969 


83453453 


20.904545 


7.588579 


191844 


84027672 


20.928449 


7.594363 


192721 


84604519 


20.952326 


7.600138 


193600 


85184000 


20.976177 


7.605905 


194481 


85766121 


21. 


7.611662 


195364 


86350388 


21.023796 


7.617411 


196249 


86938307 


21.047565 


7.623151 


197136 


87528384 


21.071307 


7.628883 


198025 


88121125 


21.095023 


7.634606 


198916 


88716536 


21.118712 


7.640321 


199809 


89314623 


21.142374 


7.646027 



SQUARES, CUBES, A:^D ROOTS. 



107 



Table — (Continued). 



Square. 



448 


200704 


89915392 


449 


201601 


90518849 


450 


202500 


91125000 


451 


203401 


91733851 


452 


204304 


92345408 


453 


205209 


92959677 


454 


206106 


93576664 


455 


207025 


94196375 


456 


207936 


94818816 


457 


208849 


95443993 


458 


209764 


96071912 


459 


210681 


96702579 


460 


211600 


97336000 


461 


212521 


97972181 


462 


213144 


98611128 


463 


214369 


99252847 


464 


215296 


99897344 


465 


216225 


100544625 


466 


217156 


101194696 


467 


218089 


101847563 


468 


219024 


102503232 


469 


219961 


103161709 


470 


220900 


103823000 


471 


221841 


104487111 


472 


222784 


105154048 


473 


223729 


105823817 


474 


224676 


106496424 


475 


225625 


107171875 


476 


226576 


107850176 


477 


227529 


108531333 


478 


228484 


109215352 


479 


229441 


109902239 


480 


230400 


13 0592000 


481 


231361 


111284641 


482 


232324 


111980168 


483 


233289 


112678587 


484 


234256 


113379904 


485 


235225 


114084125 


486 


236196 


114791256 


487 


237169 


115501303 


488 


238144 


116214272 


489 


239121 


116930169 


490 


240100 


117649000 


491 


241081 


1 J 8370771 


492 


242064 


119095488 


493 


243049 


119823157 


494 


244038 


120553784 


495 


245025 


121287375 


496 


246016 


122023936 


497 


247009 


122763473 


498 


248004 


123505992 


499 


249001 


124251499 


600 


250000 


125000000 


501 


251001 


125751501 


502 


252004 


126506008 


^03 


253009 


127263527 



Square Root. 



21.166010 

21.189620 

21.213203 

21.236760 

21.26029] 

21.283796 

21.307275 

21.330729 

21.354156 

21.377558 

21.400934 

21.424285 

21.447610 

21.470910 

21.494185 

21.517434 

21.540659 

21.563858 

21.587033 

21.610182 

21.633307 

21,656407 

21.679483 

21.702534 

21.725561 

21.748563 

21.771541 

21.794494 

21.817424 

21.840329 

21.863211 

21.886068 

21.908902 

21.931712 

21.954498 

21.977261 

22. 

22.022715 

22.045407 

22.068076 

22.090722 

22.113344 

22.135943 

22.158519 

22.181073 

22.203603 

22.226110 

22.248595 

22.271057 

22.293496 

22.315913 

22.338307 

22.360679* 

22.383029 

22.40.5356 

22.427661 



Cube Root. 



7.651725 

7.657414 

7.663094 

7.668766 

7.674430 

7.680085 

7.685732 

7.691371 

7.097002 

7.702624 

7.708238 

7.713844 

7.719442 

7.725032 

7.730614 

7.736187 

7.741753 

7.747310 

7.752860 

7.7.58402 

7.763936 

7.769462 

7.774980 

7.780490 

7.785992 

7.791487 

7.796974 

7.802453 

7.807925 

7.813389 

7.818845 

7.824294 

7.829735 

7.835168 

7.840594 

7.846013 

7.851424 

7.856828 

7.862224 

7.867613 

7.872994 

7.878368 

7.883734 

7.889094 

7.894446 

7.899791 

7.905129 

7.910460 

7.915784 

7.921100 

7.926408 

7.931710 

7.937005 

7.942293 

7.947673 

7.952847 



108 



SQUARES, CUBES, AND ROOTS. 



Table— (Continued). 



Square. 



504 


254016 


505 


255025 


506 


256036 


607 


257049 


508 


258064 


509 


259081 


510 


260100 


511 


261121 


512 


262144 


513 


263169 


514 


264196 


515 


2G5225 


516 


266256 • 


617 


267289 


518 


268324 


619 


269361 


520 


270400 


521 


271441 


522 


272484 


523 


273529 


524 


274576 


525 


275625 


526 


276676 


527 


277729 


528 


278784 


629 


279841 


530 


280900 


631 


281961 


532 


283024 


533 


284089 


534 


285156 


535 


286225 


536 


287296 


637 


288369 


538 


289444 


639 


290521 


640 


291600 


541 


292681 


642 


293764 


643 


294849 


644 


295936 


645 


297025 


646 


298116 


547 


299209 


548 


300304 


549 


301401 


650 


302500 


651 


303601 


652 


304704 


653 


305809 


564 


306916 


655 


308025 


656 


309136 


657 


310249 


658 


311364 


559 


312481 



128024064 

128787625 

129554216 

130323843 

131096512 

131872229 

132651000 

133432831 

134217728 

135005697 

135796744 

136590875 

137388096 

138188413 

138991832 

139798359 

140608000 

141420761 

142236648 

143055667 

143877824 

144703125 

145531576 

146363183 

147197952 

148035889 

148877000 

149721291 

150568768 

151419437 

152273304 

153130375 

153990656 

154854153 

155720872 

156590819 

157464000 

158340421 

159220088 

160103007 

160989184 

161878625 

162771336 

163667323 

164566592 

165469149 

166375000 

167284151 

168196608 

169112377 

170031464 

170953875 

171879616 

172808693 

173741112 

174676879 



Square Root._ 



Cube Root. 



22.449944 

22.472205 

22.494443 

22.516660 

22.538855 

22.561028 

22.583179 

22.605309 

22.627417 

22.649503 

22.671568 

22.693611 

22.715633 

22.737634 

22.759613 

22.781571 

22.803508 

22.825424 

22.847319 

22.869193 

22.891046 

22.912878 

22.934689 

22.956480 

22.978250 

23. 

23.021728 

23.043437 

23.065125 

23.086792 

23.108440 

23.130067 

23.151673 

23.173260 

23.194827 

23.216373 

23.237900 

23.259406 

23.280893 

23.302360 

23.323807 

23.345235 

23.366642 

23.388031 

23.409399 

23.430749 

23.452078 

22.473389 

23.494680 

23.515952 

23.537204 

23.558438 

23.579652 

23.600847 

23.622023 

23.643180 



7.958114 

7.963374 

7.968627 

7.973873 

7.979112 

7.984344 

7.989569 

7.994788 

8. 

8.005205 

8.010403 

8.015595 

8.020779 

8.025957 

8.031129 

8.036293 

8.041451 

8.046603 

8.051748 

8.056886 

8.062018 

8.067143 

8.072262 

8.077374 

8.082480 

8.087579 

8.092672 

8.097758 

8.102838 

8.107912 

8.112980 

8.118041 

8.12309G 

8.128144 

8.133186 

8.138223 

8.143253 

8.148276 

8.153293 

8.158304 

8.163309 

8.168308 

8.173302 

8.178289 

8.183269 

8.188244 

8.193212 

8.198175 

8.203131 

8.208082 

8.213027 

8.217965 

8.222898 

8.227825 

8.232746 

8.237661 



SQUARES, CUBES, AND ROOTS. 



109 



Table — (Continued). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


660 


313600 


175616000 


23.664319 


8.242570 


561 


314721 


17655S48I 


23.685438 


8 . 247474 


562 


315844 


177504328 


23.706539 


8.252371 


563 


316969 


178453547 


23.727621 


8 . 257263 


564 


318096 


179406144 


23.748684 


8.262149 


565 


319225 


180362125 


23.769728 


8.267029 


566 


320356 


181321496 


23.790754 


8.271903 


567 


321489 


182284263 


23.811761 


8.276772 


568 


322624 


183250432 


23.832750 


8.281635 


569 


323761 


184220009 


23.853720 


8.286493 


570 


324900 


185193000 


23.874672 


8.291344 


571 


326041 


186169411 


23.895606 


8.296190 


572 


327184 


187149248 


23.916521 


8.301030 


573 


328329 


188132517 


23.937418 


8.305865 


574 


329476 


189119224 


23.958297 


8.310694 


575 


330625 


190109375 


23.979157 


8.315517 


576 


331776 


191102976 


24. 


8.320335 


577 


332929 


192100033 


24.020824 


8.325147 


578 


334084 


193100552 


24.041630 


8.329954 


579 


335241 


194104539 


24.062418 


8.334755 


580 


336400 


195112000 


24,083189 


8.339551 


681 


337561 


196122941 


24.103941 


8-344341 


582 


338724 


197137368 


24.124676 


8.349125 


583 


339889 


198155287 


24.145392 


8.353904 


584 


341056 


199176704 


24.166091 


8.358678 


585 


342225 


200201625 


24.186773 


8.363446 


586 


343396 


201230056 


24.207436 


8.368209 


587 


344569 


202262003 


24.228082 


8.372966 


588 


345744 


203297472 


24.248711 


8.377718 


689 


346921 


204336469 


24.269322 


8 . 382465 


690 


348100 


205379000 


24.289915 


8.387206 


591 


349281 


206425071 


24.310491 


8.391942 


692 


350464 


207474688 


24.331050 


8.396673 


593 


351649 


208527857 


24.351591 


8.401398 


594 


352836 


209584584 


24.372115 


8.406118 


595 


354025 


210644875 


24.392621 


8.410832 


696 


355216 


211708736 


24.413111 


8.415541 


697 


356409 


212776173 


24.433583 


8.420245 


598 


357604 


213847192 


24.454038 


8.424944 


699 


358801 


214921799 


24.474476 


8.429638 


600 


360000 


216000000 


24.494897 


8.434327 


601 


361201 


217081801 


24.515301 


8.439009 


602 


362404 


218167208 


24.535688 


8.443687 


603 


363609 


219256227 


24.556058 


8.448360 


604 


364816 


220348864 


24.576411 


8.453027 


605 


366025 


221445125 


24.596747 


8.457689 


606 


367236 


222545016 


24.617067 


8.462347 


607 


368449 


223648543 


24.637370 


8.466999 


608 


369664 


224755712 


24.657656 


8.471647 


609 


370881 


225866529 


24.677925 


8.476289 


610 


372100 


226981000 


24.698178 


8.480926 


611 


373321 


228099131 


24.718414 


8.485557 


612 


374544 


229220928 


24.7386.33 


8.490184 


613 


375769 


230346397 


24.758836 


8.494806 


614 


376996 


231475544 


24.779023 


8.499423 


615 


378225 


232608375 
K 


24.799193 


8.5Q4034 



no 



SQUARES, CUBES, AND ROOTS. 



Table — (Continued ). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root, 


616 


379456 


233744896 


24.819347 


8.508641 


617 


380689 


234885113 


24.839484 


8.513243 


618 


381924 


236029032 


24.859605 


8.517840 


619 


383161 


237176659 


24.879710 


8.522432 


620 


384700 


238328000 


24.899799 


8.527018 


621 


385641 


239483061 


24.919871 


8.531600 


622 


386884 


240641848 


24.939927 


8.536177 


623 


388129 


241804367 


24.959967 


8.540749 


624 


389376 


242970624 


24.979992 


8.545317 


625 


390625 


244140625 


25. 


8.549879 


626 


391876 


245314376 i 


25.019992 


8.554437 


627 


393129 


246491883 


25.0.39968 


8.558990 


628 


394384 


247673152 


25.059928 


8.563537 


629 


395641 


248858189 


25.079872 


8.568080 


630- 


396900 


250047000 


25.099800 


8.572618 


631 


398161 


251239591 


25.119713 


8.577152 


632 


399424 


252435968 


25.139610 


8.581680 


633 


400689 


253636137 


25.159491 


8.586204 


634 


401956 


254S40104 


25.179356 


8.590723 


635 


403225 


256047875 


25.199206 


8.595238 


636 


404496 


257259456 


25.219040 


8.599747 


637 


405769 


258474853 


25.238858 


8.604252 


638 


407044 


259694072 


25.258661 


8.608752 


639 


408321 


260917119 


25.278449 


8.613248 


640 


409600 


262144000 


25.298221 


8.617738 


641 


410881 


263374721 


25.317977 


8.622224 


642 


412164 


264609288 


25.337718 


8.626705 


643 


413449 


265847707 


25.357444 


8.631183 


644 


414736 


267089984 


25.377155 


8.635655 


645 


416025 


268336125 


25.396850 


8.640122 


646 


417316 


269586136 


25.416530 


8.644585 


647 


418609 


270840023 


25.436194 


8.649043 


648 


419904 


272097792 


25.455844 


8.653497 


649 


421201 


273359449 


25.475478 


8.657946 


650 


422500 


274625000 


25.495007 


8.662301 


651 


423801 


275894451 


25.514701 


8.666831 


652 


425104 


277167808 


25.534290 


8.671266 


653 


426409 


278445077 


25.553864 


8.675697 


654 


427716 


279726264 


25.573423 


8.680123 


655 


429025 


281011375 


25.592967 


8 . 684545 


656 


430336 


2S2300416 


25.612496 


8.688963 


657 


431649 


283593393 


25.632011 


8.693376 


658 


432964 


284890312 


25.651510 


8.6977841 


659 


434281 


286191179 


25.670995 


8.702188^ 


660 


435600 


287496000 


25.690465 


8.706587. 


661 


436921 


288804781 


25.709920 


8.710982. 


662 


438244 


290117528 


25.720360 


8.715373 


663 


439569 


291434247 


25.748786 


8.719759 


664 


440896 


292754944 


25.768197 


8.724141 


665 


442225 


294079625 


25.787593 


8.728518- 


666 


443556 


295408296 


25.806975 


8.732891 


667 


444889 


296740963 


25.826343 


8.737200 


668 


446224 


298077632 


25.845696 


8.741624i 


669 


447561 


299418309 


25.865034 


8.7459841 


670 


448900 


300763000 


25.884358 


8.750340 


671 


450241 


302111711 


25.903667 


8.754691 



SQUARES, CUBES, AJMD ROOTS. 



Ill 



Table— (Continued). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


672 


451584 


303464448 


25.922962 


8.759038 


673 


452929 


304821217 


25.942243 


8.763380 


674 


454276 


306182024 


25.961510 


8.767719 


675 


455625 


307546875 


25.980762 


8.772053 


676 


456976 


308915776 


26. 


8.776382 


677 


458329 


310288733 


26.019223 


8.780708 


678 


459684 


311665752 


26.038433 


8.785029 


679 


461041 


313046839 


26.057628 


8.789346 


680 


462400 


314432000 


2S. 076809 


8.793659 


681 


463761 


315821241 


26.095976 


8.797967 


682 


465124 


317214568 


26.115129 


8.802272 


683 


466489 


318611987 


26.134268 


8.806572 


684 


467856 


320013504 


26.153393 


8.810868 


685 


469225 


321419125 


20.172504 


8.815159 


686 


470596 


322828856 


26.191601 


8.819447 


687 


471969 


324242703 


26.210684 


8.823730 


688 


473344 


325660672 


26.229754 


8.828009 


689 


474721 


327082769 


26.248809 


8.832285 


690 


476100 


328509000 


26.267851 


8.836556 


691 


477481 


329939371 


26.286878 


8.840822 


692 


478864 


331373888 


26.305892 


8.845085 


693 


480249 


332812557 


26.324893 


8.849344 


694 


481636 


334255384 


26.343879 


8.853598 


695 


483025 


335702375 


26.362852 


8.857849 


696 


484416 


337153536 


26.381811 


8.862095 


697 


485809 


338608873 


26.400757 


8.866337 


698 


487204 


340068392 


26.419689 


8.870575 


699 


488601 


341532099 


26.433608 


8.874809 


700 


490000 


343000000 


26.457513 


8.879040 


701 


491401 


344472101 


26.476404 


8.883266 


702 


492804 


345948088 


26.495282 


8.887488 


703 


494209 


347428927 


26.514147 


8.891706 


704 


495616 


348913664 


26.532998 


8.895920 


705 


497025 


350402625 


26.551836 


8.900130 


706 


498436 


351895816 


28.570660 


8.904336 


707 


499849 


353393243 


26.589471 


8.908538 


708 


501264 


354894912 


26.608269 


8.912736 


709 


502681 


356400829 


26.627053 


8.916931 


710 


504100 


357911000 


26.645825 


8.921121 


711 


505521 


359425431 


26.664583 


8.925307 


712 


506944 


360944128 


26.683328 


8.929490 


.713 


508369 


362467097 


26.702059 


8.933668 


714 


509796 


363994344 


26.720778 


8.937843 


715 


511225 


365525875 


26.739483 


8.942014 


716 


512656 


367061696 


26.758176 


8.946180 


717 


514089 


368601813 


26.776855 


8.950343 


718 


515524 


370146232 


26.795522 


8.954502 


719 


516961 


371694959 


26.814175 


8.958658 


720 


518400 


373248000 


26.832815 


8.962809 


721 


519841 


374805361 


26.851443 


8.966957 


722 


521284 


376367048 


26.870057 


8.971100 


723 


522729 


377933067 


26.888659 


8.975240 


724 


524176 


379503424 


26.907248 


8.979376 


725 


525625 


381078125 


26.925824 


8.983508 


726 


527076 


382657176 


26.944387 


8.987637 


727 


528529 


384240583 


26.962937 


8.991762 



112 



SQUARES, CUBES, A2iD ROOTS. 



Table — (Continued). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Roof. 


728 


529984 


385828352 


26.981475 


8.995883 


729 


531441 


387420489 


27. 


9. 


730 


532900 


389017000 


27.018512 


9.004113 


731 


534361 


390617891 


27.037011 


9.008222 


732 


535824 


392223168 


27.055498 


9.012328 


733 


537289 


393832837 


27.073972 


9.0164S0 


734 


538756 


395446904 


27.092434 


9.020529 


735 


540225 


397065375 


27.110883 


9.024623 


736 


541696 


3986SS256 


27.129319 


9.028714 


737 


543169 


400315553 


27.147743 


9.032802 


738 


544644 


401947272 


27.166155 


9.036885 


739 


546121 


403583419 


27.184554 


9.040965 


740 


547600 


405224000 


27.202941 


9.045041 


741 


549081 


406869021 


27.221315 


9.049114 


742 


550564 


408518488 


27.239676 


9.053183 


743 


552049 


410172407 


27.258026 


9.057248 


744 


553536 


411830784 


27.276363 


9.061309 


745 


555025 


413493625 


27.294688 


9.065367 


746 


556516 


415160936 


27.313000 


9.069422 


747 


558009 


416832723 


27.331300 


9.073472 


748 


559504 


41S50S992 


27.349588 


9.077519 


749 


561001 


420189749 


27.367864 


9.081563 


750 


562500 


421875000 


27.386127 


9.085603 


751 


564001 


423564751 


27.404379 


9.089639 


752 


565504 


425259008 


27.422618 


9.093672 


753 


567009 


42-6957777 


27.440845 


9.097701 


754 


568516 


428661064 


27.459060 


9.101726 


755 


570025 


430368375 


27.477263 


9.105748 


756 


571536 


432081216 


27.495454 


9.109766 


757 


573049 


433798093 


27.513633 


9.113781 


758 


574564 


435519512 


27.531799 


9.117793 


759 


576081 


437245479 


27.549954 


9.121801 


760 


577600 


438976000 


27.568097 


9.125805 


761 


579121 


440711081 


27.586228 


9.129806 


762 


580644 


442450728 


27.604347 


9.133803 


763 


582169 


444194947 


27.622454 


9.137797 


764 


583696 


445943744 


27.640549 


9.141788 


765 


585225 


447697125 


27.658633 


9.145774 


766 


586756 


449455096 


27.676705 


9.149757 


767 


588289 


451217663 


27.694764 


9.153737 


768 


589824 


452984832 


27.712812 


9.157713 


769 


591361 


454756609 


27.730849 


9.161686 


770 


592900 


456533000 


27.748873 


9.165656 


771 


594441 


458314011 


27.766886 


9.169622 


772 


595984 


460099648 


27.784888 


9.173585 


773 


597529 


461889917 


27.802877 


9.177544 


774 


599076 


463684824 


27.820855 


9.181500 


775 


600625 


465484375 


27.838821 


9.185452 


776 


602 176 


467288576 


27.856776 


9.189401 


77.7 


603729 


469097433 


27.874719 


9.193347 


778 


605284 


470910952 


27.892651 


9.197289 


779 


606841 


472729139 


27.910571 


9.201228 


780 


608400 


474552000 


27.928480 


9.205164 


781 


609961 


476379541 


27.946377 


9.209096 


782 


611524 


478211768 


27.964262 


9.213025 


783 


613089 


480048687 


27.982137 


9.216950 



SQUARES, CUBES, AND ROOTS. 



113 



Table — (Continued). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root 


784 


614656 


4^1890304 


28. 


9.220872 


785 


616225 


483736025 


28.017851 


9.224791 


786 


617796 


485587656 


28.035691 


9.228706 


787 


619369 


487443403 


28.053520 


9.232618 


788 


620944 


489303872 


28.071337 


9.237527 


789 


622521 


491169069 


28.089143 


9.240433 


790 


624100 


493039000 


28.106938 


9.244335 


791 


625681 


494913671 


28.124722 


9.248234 


792 


627264 


496793088 


28.142494 


9.252130 


793 


628849 


498677257 


28.160255 


9.256022 


794 


630436 


500566184 


28.178005 


9.2.^9911 


795 


632025 


502459875 


28.195744 


9.263797 


796 


633616 


504358336 


28.213472 


9.267679 


797 


635209 


506261573 


28.231188 


9.271559 


798 


636804 


508169592 


28.248893 


9.275435 


799 


638401 


510082399 


28.266588 


9.279308 


800 


640000 


512000000 


28.284271 


9.283177 


801 


641601 


513922401 


28.301943 


9.287044 


802 


643204 


515849608 


28.319604 


9.290907 


803 


644809 


517781627 


28.337254 


9.294767 


804 


646416 


519718464 


28.354893 


9.298623 


805 


648025 


521660125 


28.372521 


9.302477 


806 


649636 


523606616 


28.390139 


9.308327 


807 


651249 


525557943 


28.407745 


9.310175 


808 


652864 


527514112 


28.425340 


9.314019 


809 


654481 


529475129 


28.442925 


9.317859 


810 


656100 


531441000 


28.460498 


9.321697 


811 


657721 


533411731 


28.478061 


9.325532 


812 


659344 


535387328 


28.495613 


9.329363 


813 


660969 


537366797 


28.513154 


9.333191 


814 


662596 


539353144 


28.530685 


9.337016 


815 


664225 


541343375 


28.548204 


9.340838 


816 


665856 


543338496 


28.565713 


9.344657 


817 


667489 


545338513 


28.583211 


9.348473 


818 


669124 


547343432 


28.600699 


9.352285 


819 


670761 


549353259 


28.618176 


9.356095 


820 


672400 


551368000 


28.635642 


9.359901 


821 


674041 


553387661 


28.653097 


9.363704 


822 


675684 


555412248 


28.670542 


, 9.367505 


823 


677329 


557441767 


28.687976 


9.371302 


824 


678976 


559476224 


28.705400 


9.375096 


825 


680625 


561515625 


28.722813 


9.378887 


826 


682276 


563559976 


28.740215 


9.372675 


827 


683929 


565609283 


28.757607 


9.386460 


828 


685584 


567663552 


28.774989 


9.390241 


829 


687241 


569722789 


28.792360 


9.394020 


830 


688900 


571787000 


28.809720 


9.397796 


831 


690561 


673856191 


28.827070 


9.401569 


832 


692224 


575930368 


28.844410 


9.405338 


833 


693889 


578009537 


28.861739 


9.409105 


834 


695556 


580093704 


28.879058 


9.412869 


835 


697225 


582182875 


28.896366 


9.416630 


83G 


698896 


584277056 


28.913664 


9.420387 


837 


700569 


586376253 


28.930952 


9.424141 


838 


702244 


588480472 


28.948229 


9.427893 


839 


703921 


590589719 
K2 


28.965496 


9.431642 



tu 



SQUARES, CUBES, AND ROOTS. 

Table— (Continued). 



Number. 


Square. 


Cube. 


Square Root. 


840 


705600 


592704000 


28.982753 


841 


707281 


594823321 


29. 


842 


708964 


596947688 


29.017236 


843 


710649 


599077107 


29.034462 


844 


712336 


601211584 


29.051678 


845 


714025 


603351125 


29.068883 


846 


715716 


605495736 


29.086079 


847 


717409 


607645423 


29.103264 


848 


719104 


609800192 


29.120439 


849 


720801 


611960049 


29.137604 


850 


722500 


614125000 


29.154759 


851 


724201 


616295051 


29.171904 


852 


725904 


618470208 


29.189039 


853 


727609 


620650477 


29.206163 


854 


729316 


622835864 


29.223278 


855 


731025 


625026375 


29.240383 


856 


732736 


627222016 


29.257477 


857 


734449 


629422793 


29.274562 


858 


736164 


631628712 


29.291637 


859 


737881 


633839779 


29.308701 


860 


739600 


636056000 


29.325756 


861 


741321 


638277381 


29.342801 


862 


743044 


640503928 


29.359836 


863 


744769 


642735647 


29.376861 


864 


746496 


644972544 


29.393876 


865 


748225 


647214625 


29.410882 


866 


749956 


649461896 


29.427877 


867 


751689 


651714363 


29.444863 


868 


753424 


653972032 


29.461839 


869 


755161 


656234909 


29.478805 


870 


756900 


658503000 


29.495762 


871 


758641 


660776311 


29.512709 


872 


760384 


663054848 


29.529646 


873 


762129 


665338617 


29.546573 


874 


763876 


667627624 


29.563491 


875 


765625 


669921875 


29.580398 


876 


767376 


672221376 


29.597297 


877 


769129 


674526133 


29.614185 


878 


770884 


676836152 


29.631064 


879 


772641 


679151439 


29.647932 


880 


774400 


681472000 


29.664793 


881 


776161 


683797841 


29.681644 


882 


777924 


686128968 


29.698484 


883 


779689 


688465387 


29.715315 


884 


781456 


690807104 


29.732137 


885 


783225 


693154125 


S9. 748949 


886 


784996 


695506456 


29.765752 


887 


786769 


697864103 


29.782545 


888 


788544 


700227072 


29.799328 


889 


790321 


702595369 


29.816103 


890 


792100 


704969000 


29.832867 


891 


793881 


707347971 


29.849623 


892 


795664 


709732288 


29.866369 


893 


797449 


712121957 


29.883105 


894 


799236 


714516984 


29.899832 


895 


801025 


716917375 


1 29.916550 ! 



SQUARES, CUBES, AND ROOTS. 
Table — (Continued). 



115 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


896 


802816 


719323136 


29.933259 


9.640569 


897 


804609 


721734273 


29.949958 


9.644154 


898 


806404 


724150792 


29.966648 


9.647736 


899 


808201 


726572699 


29.983328 


9.651316 


900 


810000 


729000000 


30. 


9.654893 


901 


811804 


731432701 


30.016662 


9.658468 


902 


813604 


733870808 


30.033314 


9.662040 


903 


815409 


736314327 


30.049958 


9.665609 


904 


817216 


738763264 


30.066592 


9.669176 


905 


819025 


741217625 


30.083217 


9.672740 


906 


820836 


743677416 


30.099833 


9.676301 


907 


822649 


746142643 


30.116440 


9.679860 


908 


824464 


748613312 


30.133038 


9.683416 


909 


826281 


751089429 


30.149626 


9.686970 


910 


828100 


753571000 


30.166206 


9.690521 


911 


829921 


756058031 


30.182776 


9.694069 


912 


831744 


758550528 


30.199337 


9.697615 


913 


833569 


761048497 


30.215889 


9.701158 


914 


835396 


763551944 


30.232432 


9.704698 


915 


837225 


766060875 


30.24S966 


9.708236 


916 


839056 


768575296 


30.265491 


9.711772 


917 


840889 


771095213 


30.282007 


9.715305 


918 


842724 


773620632 


30.298514 


9.718835 


919 


844561 


776151559 


30.315012 


9.722363 


920 


846400 


778688000 


30.331.501 


9.725888 


921 


848241 


781229961 


30.347981 * 


9.729410 


922 


850084 


783777448 


30.364452 


9.732930 


923 


851929 


786330467 


30.380915 


9.736448 


924 


853776 


788889024 


30.397368 


9.739963 


925 


855625 


791453125 


30.413812 


9.743475 


926 


857476 


794022776 


30.430248 


9.746985 


927 


859329 


796597983 


30.446674 


9.7.50493 


928 


861184 


799178752 


30.463092 


9.753998 


929 


863041 


801765089 


30.479501 


9.757500 


930 


864900 


804357000 


30.495901 


9.761000 


931 


866761 


806954491 


30.512292 


9.764497 


932 


868624 


809557568 


30.528675 


9.767992 


933 


870489 


8121662S7 


30.545048 


9.771484 


934 


872356 


814780504 


30.561413 


9.774974 


935 


874225 


817400375 


30.577769 


9.778461 


936 


876096 


820025856 


30.594117 


9.782946 


937 


877969 


822656953 


30.610455 


9.785428 


938 


879844 


825293672 


30.626785 


9.788908 


939 


881721 


827936019 


30.643106 


9.792386 


940 


883600 


830584000 


30.659419 


9.795861 


94] 


885481 


833237621 


30.675723 


9.799333 


942 


887364 


835896888 


30.692018 


9.802803 


943 


889249 


838561807 


30.708305 


9.806271 


944 


891136 


841232384 


30.724583 


9.809736 


945 


893025 


843908625 


30.740852 


9.813198 


946 


894916 


846590536 


30.757113 


9.816659 


947 


896809 


849278123 


30.773365 


9.820117 


948 


898704 


851971392 


30.789608 


9.823572 


949 


900601 


854670349 


30.805843 


9.827025 


950 


902500 


857375000 


30.822070 


9.830475 


951 


904401 


8600S5351 


30.838287 


9.833923 



116 



SqUAEES, CUBES, AND ROOTS. 







Table— (Continued). 




Number. 


Sqaure. 


Cube. 


Square Root. 


Cube Root. 


952 


906304 


862801408 


30.854497 


9.837369 


953 


908209 


865523177 


30.870698 


9.840812 


954 


910116 


868250664 


30.886890 


9.844253 


955 


912025 


870983875 


30.903074 


9.847692 


956 


913936 


873722816 


30.919249 


9.851128 


957 


915849 


876467493 


30.935416 


9.854561 


958 


917764 


879217912 


30.951575 


9.857992 


959 


919681 


881974079 


30.967725 


9.861421 


960 


921600 


884736000 


30.983866 


9.864848 


961 


923521 


887503681 


31. 


9.868272 


962 


925444 


890277128 


31.016124 


9.871694 


963 


927369 


893056347 


31.032241 


9.875113 


964 


929296 


895841344 


31.048349 


9.878530 


965 


931225 


898632125 


31.064449 


9.881945 


966 


933156 


9014*^8696 


31.080540 


9.885357 


967 


935089 


90423 J 063 


31.096623 


9.888767 


968 


937024 


907039232 


31.112698 


9.892174 


969 


938961 


909853209 


31.128764 


9.895580 


970 


940900 


912673000 


31.144823 


9.898983 


971 


942841 


915498611 


31.160872 


9.902383 


972 


944784 


918330048 


31.176914 


9.905781 


973 


946729 


921167317 


31.192947 


9.909177 


974 


948676 


924010424 


31.208973 


9.9J2571 


975 


950625 


926859375 


31.224990 


9.915962 


976 


95257J5 


929714176 


31.240998 


9.919351 


977 


954529 


932574833 


31.256999 


9.922738 


978 


956484 


935441352 


31.272991 


9.926122 


979 


958441 


938313739 


31.288975 


9.929504 


980 


960400 


941192000 


31.304951 


9.932883 


981 


962361 


944076141 


31.320919 


9.936261 


982 


964324 


946966168 


31.336879 


9.939636 


983 


966289 


949862087 


31.352830 


9.943009 


984 


968256 


952763904 


31.368774 


9.946379 


985 


970225 


955671625 


31.384709 


9.949747 


986 


972196 


958585256 


31.400636 


9.953113 


987 


974169 


961504803 


31.416556 


9.956477 


988 


976144 


964430272 


31.432467 


9.959839 


989 


978121 


967361669 


31.448370 


9.963198 


990 


980100 


970299000 


61.464265 


9.966554 


991 


982081 


973242271 


31.480152 


9.969909 


992 


984064 


976191488 


31.496031 


9.973262 


993 


986049 


979146657 


31.511902 


9.976612 


994 


988036 


982107784 


31.527765 


9.979959 


995 


990025 


985074875 


31.543620 


9.983304 


996 


992016 


988047936 


31.559467 


9.986648 


997 


994009 


991026973 


31.575306 


9.989990 


998 


996004 


994011992 


31.591138 


9.993328 


999 


998001 


997002999 


31.606961 


9.996665 


1000 


1000000 


1000000000 


31.622776 


10. 



Additional use of this table can be made by the aid of the following Rules : 

To find the Square of a Number above 1000 =^when the Number 
is divisible by any Number without leaving a Remainder. 
Rule. — If the number exceed by 2, 3, or any other number of times, any 



SQUARES, CUBES, AND ROOTS. 117 

number contained in the table, let the square affixed to that number in the table 
be multiplied by the square of 2, 3, 4, 5, or 6, &c., and the product will be the 
answer. 

Example. — Required the square of 1550. 

1550 is 10 times 155, and the square of 155 in the table is 24025. 
Then 24025X10^ = 2402500 Ans. 

When the Number is an Odd Number. 

Rule.— Find the two numbers nearest to each other, which, added together, 
make that sum ; then the sum of the squares of these two numbers, as per table, 
multiplied by 2, will give the answer, exceeded by 1, which is to be subtracted, and 
the remainder is the answer required. 

Example.— What is the square of 1345 1 

The nearest two numbers are j g 'g | = 1345. 

Then ner table \ 6732 = 452929 
men, per table, j 6722- 452584 

904513X2 =: 1809026—1 = 1809025 Ans. 

To find the Cube of a Number greater than is contained in the 

Table. 

Rule.— Proceed as in squares to find how many times the number exceeds one 
of the tabular numbers. Multiply the cube of that number by the cube of the 
number of times the number sought exceeds the number in the table, and the prod- 
uct will be the answer. 

Example.— What is the cube of 1200 1 

1200 is 3 times 400, and the cube of 400 is 64000000. 

Then 64000000X3^ = 1728000000 Ans. 

To find the Squares of Numbers following each other m Arith- 
metical Progression. 

Rule.— Find the squares of the two first numbers in the usual way, and subtract 
the less from the greater. Add the difference to the greatest square, with the ad- 
dition of 2 as a constant quantity ; the sum will be the square of the next 
number. 

Example.— What are the squares of 1001, 1002, 1003, 1004, and 1005 '» 
10002 = 1000000 
9992= 998001 

1999 
Add . . 2 

Add 10002 = 1000000 

1002001 Square of 1001. 
Difference, 2001+2= 2003 

1004004 Square of 1002. 
Difference, 2003+2= 2005 

1006009 Square of 1003. 
Difference, 2005+2 = 2007 

1008016 Square of 1004. 
Difference, 2007+2= 2009 

1010025 Square of 1005. 

To find the Cubes of Numbers following each other in Arithmeti- 
cal Progression. 

Rule.— Find the cubes of the two first numbers, and subtract the less from the 
greater ; then multiply the least of the two numbers cubed, by 6; add the product, 



118 SQUARES, CUBES, AND ROOTS. 

with the addition of 6, to the difference, and continue this the first series of diffe^ 
ences. 

For the second series of differences, add the cube of the highest of the above 
numbers to the difference, and the sum will be the cube of the next number. 

Example.— What are the cubes of 1001, 1002, and loos'? 

First Series. 
Cube of 1000 = 1000000000 

Cube of 999 = 997002999 a 

2997001 Difference. 
999x6+6 = 6000 

3003001 Difference of 1000. 
6000 +6 = 6008 

"3009007 Difference of 1001. 

6006 +6 = 6012 

3015019 Difference of 1002. 

Second Series. 
Cube of 1000 . =1000000000 
Difference for 1000, 3003001 

1003003001 = Cube of 1001. 
Difference for 1001, 3009007 

1006012008 = Cube of 1002. 
Difference for 1002, 3015019 

1009027027 = Cube of 1003. 

To find the Cube or Square Root of a higher Number than %s 
contained in the Table. 

Rule.— Find in the column of Squares or Cubes the number nearest to that num- 
ber whose root is required, and the number from which that square or cube is de- 
rived will be the answer when decimals are not of importance. 

Example.— What is the square root of 562500 ? 

In the table of Squares, this nmnber is the square of 750 ; therefore 7o0 is the 
square root required. 

Example.— What is the cube root of 2248090 1 

In the table of Cubes, 2248091 is the cube of 131 ; therefore 131— is the cube 
root required, nearly. 

To find the Cube Root of any Number over 1000. 

Rule.— Find by the table the nearest cube to the number given, and call it the 
assumed cube. Multiply the assumed cube and the given number respectively by 
2 ; to the product of the assumed cube add the given number, and to the product 
of the given number add the assumed cube. 

Then, as the sum of the assumed cube is to the sum of the given number, so is 
the root of the assumed cube to the root of the given number. 

Example.— What is the cube root of 224809 ? 

By table, the nearest cube is 216000, and its root is 60. 

216000x2+224809 = 656809, 
And 224809x2+216000 = 665618. 
Then, as 656809 : 665618 : : 60 : 60.804+ 

To find the Sixth Root of a Number. 

Rule.— Take the cube root of its square root. 
Example.— What is the ^ of 441 ? 

^441 = 21 , and .^21 = 2.7589 ^ns. 



SQUARES, CUBES, AND ROOTS. 



119 



TO FIND THE CUBE OR SQUARE ROOT OF A NUMBER CONSIST- 
ING OF INTEGERS AND DECIMALS. 

Rule.— Multiply the difference between the root of the integer part and the root 
of the next higher integer by the decimal, and add the product to the root of the 
integer given ; the sum will be the root of the number required. 

This is correct for the square root to three places of decimals, and in the cube root 
to seven. 

Example.— What is the square root of 53.75, 
V' 54 = 7.3484 



y/ .53= 7.2801 



.051225 
53 = 7.2801 
V'53.75 = 7.331325 



^/ 



, and the cube root of 843.75 1 
^844 = 9.4503 



3/843 = 9.4466 
.0037 

^ 

.002775 

^843 = 9.4466 

4^843.75 = 9.449375 



120 

Table 



SIDES OF EQUAL SQUARES. 

of the Sides of Squares^equal in Area to 
Diameter, from 1 to 100. 



a Circle of any 



Side of equal 
Square. 



Diameter. 



Side of equal 
Square. 



0.886 
1.107 
1.329 
1.550 
1.772 
1.994 
2.215 
2.437 
2.658 
2.880 
3.101 
3.323 
3.544 
3.766 
3.988 
4.209 
4.431 
4.652 
4.874 
5.095 
5.317 
6.538 
5.760 
5.982 
6.203 
6.425 
6.646 
6.868 
7.089 
7.311 
7.532 
7.754 
7.976 
8.197 
8.419 
8.640 
8.862 
9.083 
9.305 
9.526 
9.748 
9.970 
10.191 
10.413 
10.634 
10.856 
11.077 
11.299 
11.520 
11.742 
11.964 
12.185 
12.407 
12.628 
12. '850 
13.071 



15. 
.25 
.5 
.75 
16. 
.25 
.5 
.75 
17. 
.25 
.5 
.75 
18. 
.25 
.5 
.75 
19. 
.25 
.5 
.75 
20. 
.25 
.5 
.75 
21. 
.25 
.5 
.75 
22. 
.25 
.5 
.75 
23. 
.25 
.5 
.75 
24. 
' .25 
.5 
.75 
25. 
.25 
.5 
.75 
26. 
.25 
.5 
.75 
27. 
.25 
.5 
.75 
28. 
.25 
.5 
.75 



13.293 

13.514 

13.736 

13.958 

14.179 

14.401 

14.622 

14.844 

15.065 

15.287 

15.508 

15.730 

15.952 

16.173 

16.395 

16.616 

16.838 

17.059 

17.281 

17.502 

17.724 

17.946 

18.167 

18.389 

18.610 

18.832 

19.053 

19.275 

19.496 

19.718 

19.940 

20.161 

20.383 

20.604 

20.826 

21.047 

21.269 

21.491 

21.712 

21.934 

22.155 

22.377 

22.598 

22.820 

23.041 

23.263 

23.485 

23.706 

23.928 

24.149 

24.371 

24.592 

24.814 

25.035 

25.257 

25.479 



Diameter. 

"297" 



Side of equal 
Squire. 



.25 
.5 
.75 
30. 
.25 
.5 
.75 
31. 
.25 
.5 
.75 
32. 
.25 
.5 
.75 
33. 
.25 
.5 
.75 
34. 
.25 
.5 
.75 
35. 
.25 
o5 
.75 
36. 
.25 
.5 
.75 
37. 
.25 
.5 
.75 
38. 
.25 
.5 
.75 
39. 
.25 
.5 
.75 
40. 
.25 
.5 
.75 
41. 
.25 



Diameter. 



42. 



.75 

.25 

.5 

.75 



25.700 

25.922 

26.143 

26.365 

26.586 

26.808 

27.029 

27.251 

27.473 

27.694 

27.916 

28.137 

28.359 

28.580 

28.802 

29.023 

29.245 

29.467 

29.688 

29.910 

30.131 

30.353 

30.574 

30.796 

31.017 

31.239 

31.461 

31.682 

31.904 

32.125 

32.347 

32.568 

32.790 

33.011 

33.233 

33.455 

33.676 

33.898 

34.119 

34.341 

34.562 

34.784 

35.005 

35.227 

35.449 

35.670 

35.892 

36.113 

36.335 

36.556 

36.778 

36.999 

37.221 

37.443 

37.664 

37.886 



43. 
.25 
.5 
.75 
44. 
.25 
.5 
.75 
45. 
.25 
.5 
.75 
46. 
.25 
.5 
.75 
47. 
.25 
.5 
.75 
48. 
.25 
.5 
.75 
49. 
.25 
.5 
.75 
50, 
.25 
.5 
.75 
51. 
.25 
.5 
.75 
52. 
.25 
.5 
.75 
53. 
.25 
.5 
.75 
54. 
.25 
.5 
.75 
55. 
.25 
.5 
.75 
56. 
.25 
.5 
.75 



{Side of equal 
j Square. 

38.107 

38.329 

38.550 

38.772 

38.993 

39.215 

39.437 

39.658 

39.880 

40.101 

40.323 

40.544 

40.766 

40.987 

41.209 

41.431 

41.652 

41.874 

42.095 

42.317 

42.538 

42.760 

42.982 

43.203 

43.425 

43.646 

43.868 

44.089 

44.311 

44.532 

44.754 

44.976 

45.197 

45.419- 

45.640 

45.862 

46.083 

46.505 

46.526 

46.748 

46.970 

47.191 

47.413 

47.634 

47.856 

48.077 

48.299 

48.520 

48.742 

48.964 

49.185 

49.407 

49.628 

49.850 

.50.071 

50 .293 



SIDES OF EQUAL SQUARES. 
Table— (Continued). 



121 



Diameter. 


Side of equal 
Square. 


Diameter. 


Side of equal 
S^qiiare. 


Diameter. 


Side of equal 
Square. 


Diameter. 


Side of equal 
Square 


67. 


50.514 


68. 


60.263 


79. 


70.011 


90. 


79.760 


.25 


50.736 


.25 


60.484 


.25 


70.233 


.25 


79.981 


.5 


50.958 


.5 


60.706 


.5 


70.455 


.5 


80.203 


.75 


51.179 


.75 


60.928 


.75 


70.676 


.75 


80.425 


68. 


51.401 


69. 


61.149 


80. 


70.898 


91. 


80.646 


.25 


51.622 


.25 


61.371 


.25 


71.119 


.25 


80.868 


.5 


51.844 


.5 


61.592 


.5 


71.341 


.5 


81.089 


.75 


52.065 


.75 


61.814 


.75 


71.562 


.75 


81.311. 


69. 


52.287 


70. 


62.035 


81. 


71.784 


92. 


81.532 


.25 


52.508 


.25 


62.257 


.25 


72.005 


.25 


81.754 


.5 


52.730 


.5 


62.478 


.5 


72.227 


.5 


81.975 


.75 


52.952 


.75 


62.700 


.75 


72.449 


.75 


82.197 


60. 


53.173 


71. 


62.922 


82. 


72.670 


93. 


82.419 


.25 


53.395 


.25 


63.143 


.25 


72.892 


.25 


82.640 


.5 


53.616 


.5 


63.365 


.5 


73.113 


.5 


82.862 


.75 


53.838 


.75 


63.586 


.75 


73.335 


.75 


83.083 


61 


54.059 


72. 


63.808 


83. 


73.556 


94. 


83.305 


.25 


54.281 


.25 


64.029 


.25 


73.778 


.25 


83.526 


.5 


54.502 


.5 


64.251 


.5 


73.999 


.5 


83.748 


.75 


54.724 


.75 


64.473 


.75 


74.221 


.75 


83.970 


62. 


54.946 


73. 


64.694 


84. 


74.443 


95. 


84.191 


.25 


55.167 


.25 


64.916 


.25 


74.664 


.25 


84.413 


.5 


55.389 


.5 


65.137 


.5 


74.886 


.5 


84.634 


.75 


55.610 


.75 


65.359 


.75 


75.107 


.75 


84.856 


63. 


55.832 


74. 


65.580 


85. 


75.329 


96. 


85.077 


.25 


56.053 


.25 


65.802 


.25 


75.550 


.25 


85.299 


.5 


56.275 


.5 


66.023 


.5 


75.772 


.5 


85.520 


.75 


56.496 


.75 


66.245 


.75 


75.993 


.75 


85.742 


64. 


56.718 


75. 


66.467 


86. 


76.215 


97. 


85.964 


.25 


56.940 


.25 


66.688 


.25 


76.437 


.25 


86.185 


.5 


57.161 


.5 


66.910 


.5 


76.658 


.5 


86.407 


.75 


57.383 


.75 


67.191 


.75 


76.880 


.75 


86.628 


65. 


57.604 


76. 


67.353 


87. 


77.101 


98. 


86.850 


.25 


57.826 


.25 


67.574 


.25 


77.323 


.25 


87.071 


.5 


58.047 


.5 


67.796 


.5 


77.544 


.5 


87.293 


.75 


58.269 


.75 


68.017 


.75 


77.766 


.75 


87.514 


66. 


58.490 


77. 


68.239 


88. 


77.987 


99. 


87.736 


.25 


58.712 


.25 


68.461 


.25 


78.209 


.25 


87.958 


.5 


58.934 


.5 


68.682 


.5 


78.431 


.5 


88.179 


.75 


59.155 


.75 


68.904 


.75 


78.652 


.75 


88.401 


67. 


59.377 


78. 


69.125 


89. 


78.874 


100. 


88.622 


.25 


59.598 


.25 


69.347 


.25 


79.095 


.25 


88.844 


.5 


59.820 


.5 


69.568 


.5 


79.317 


.5 


89.065 


.75 


60.041 


.75 


69.790 


.75 


79.538 


.75 


89.287 



USE OF THIS TABLE. 

To find a Square that shall have the same Area as a Given Circle. 

Example.— What is the side of a square that has the same area as a circle of 
73i inches ? .^, . , j . .^ x. 

By table of Areas, page 93, opposite to 73.25 is its area, 4214.1 ; and in the above 
table, page 121, is 64.916, the side of a square that has the same area as a circle of 
73^ inches in diameter. 

Example.— What should be the side of a square that would give the same area 
as a board that is 18 inches v^de and 10 feet long 1 • 



122 SIDES OF EQUAL SQUARES. 

18 inches is . 1.5 feet. 

^10 

15.0 feet. 

14 4 square inches in a foot. 

60F 
600 
150 

2160.0 inches area. 
Bv table page 120, 2164.75 inches area have a diameter of 52.5 inches, which in 
the above table gives an equal side of 46.526, which is the answer very nearly. 



PLANE TRIGONOMETRY. 

ABC the three angles (A the right angle) ; a Z»c the three sides respectively o]^ 
posite to them ; R the tabular radius (1 or 1000000) ; S the area of the triangle, and 

y half its perimeter = ( — ^ — )' 



RIGHT-ANGLED TRIANGLES. 




QCasinth. 



E C -.^ 




, also = a. - 



Given. To find A C and B A. 

Hyp. B C, J R : B C : : sin. B : A C, 
and Angles, j R : B C : : sin. C : B A. 

61VEN. To find B A and B C. 

AC, S R : AC : : tan. C : B A, 
and Angles. ^ R : A C : : sec. C : B C, 
J or sin. B . AC : : R : BC. 

Given. To find Angles and A C. 

Hyp. B C, < B C : R : : B A : sin. C, whose comp. is B. 
and leg B A. ^ R : B C : : sin. B : A C. 

Given. 

Both legs. I 



sin. B 



To find Angles and B C. 
A C : R : : B A : tan. C, whose comp. is B. 



Note. 



1 fein. C : B A : : R : B C, 
I or R : AC : : sec. C : BC. 
-By sin. or tan. B or C is meant the sine or tangent of the angle B or C. 



Let A B C be a right-angled triangle, in which A B is as- 
sumed to be radius ; B C is the tangent of A, and A C its 
secant to that radius ; or, dividing each of these by the base, 
we shall have the tangent and secant of A respectively to 
radius 1. Tracing the consequences of assuming B C and 
A C each for radius, we obtain the following expressions : 




-Jb 



— — = tan. angle A. 
base 

-^ = sec. angle A. 
base 

P— ^* = sin. angle A. 
hyp. 



base 
perp. 
hyp. 
perp. 
base 
hyp. 



= tan. angle C. 
= sec. angle C 
= sin. angle C. 



124 



PLANE TRIGONOMETRY. 



OBLIQUE-ANGLED TRIANGLES. 





-A, G . C D A D / ^ O 

Given, the Angles and Side A B, to find B C and A C. 



Sin. C 


AB : 


: sin. A 


BC. 


Sin. C 


AB : 


: sin. B 


AC. 



Given, two Sides A B, B C, and the Angle C, to find Angle A and B, 
and Side A C . 

A B : sin. C : : B C : sin. A, which, added to C, and the sum subtracted from 
180O, will give B. 

Sin. C : A B : : sin. E : A C. 

Given, A C, AB, and the included Angle A, to find Angles C and B, 
and Side B C. 

Subtract half the given angle A from 90° ; the remainder is half the %um of the 
other angles. Then, as the sum of the sides A C, A B is to their difference, so is 
the tangent of the half sum of the other angles to the tangent of half their differ- 
ence, which, added to and subtracted from the half sum, will give the two angles 
B and C, the greatest angle being opposite to the greatest side. 
Sin. B : A C : : sin. A : B C. 

Given, all three Sides, to find all the Angles. 

Let fall a perpendicular B D opposite to the required angle ; then, as A C : sum 
of A B, B C : : their difference : twice D G, the distance of the perpendicular from 
the middle of the base ; hence A D, C D are known, and the triangle A B C is divi- 
ded mto two right-angled tiiangles B C D, B A D ; then, by the rules in right-angled 
triangles, find the angle A or C. *«» & 



NATCKAL SINES, COSINES, AND TANGENTS. 



125 





Table 


of Natural Sines 


Cosines 


, and Tangents. 




D.M. 


Sine. 


Cosine. 


Tangent. 


D.M 


, Sine. 


Cosine. 


Tangent. 


.15 


00436 


99999 


00436 


14. 


24192 


97030 


24933 


.30 


00872 


99996 


00873 


.15 
.30 


24615 
25038 


96923 
96815 


25397 
25862 


.45 


01309 


99991 


01309 


.45 


25460 


96705 


26328 


1. 


01745 


99985 


01745 


15. 


25882 


96593 


26794 


.15 


02181 


99976 


02182 


.15 


26303 


96479 


27263 


.30 


02618 


99966 


02619 


.30 


26724 


96363 


27732 


.45 


03054 


99953 


03055 


.45 


27144 


96246 


28203 


2. 


03490 


99939 


03492 


16. 


27564 


96126 


28675 


.15 


03926 


99923 


03929 


.15 


27983 


96005 


29147 


.30 


04362 


99905 


04366 


.30 


28402 


95882 


29621 


.45 


04798 


99885 


04Q03 


.45 


28820 


95757 


30097 


3. 


05234 


99863 


05241 


17. 


29237 


95630 


30573 


.15 


05669 


99839 


05678 


.15 


29654 


95502 


31051 


.30 


06105 


99813 


06116 


.30 


30071 


95372 


31530 


.45 


06540 


99786 


06554 


.45 


30486 


95240 


32010 


4. 


06976 


99756 


06993 


18. 


30902 


95106 


32492 


.15 


07411 


99725 


07431 


.15 


31316 


94970 


32975 


.30 


07846 


99692 


07870 


.30 


31730 


94832 


33460 


.45 


08281 


99657 


08309 


.45 


32144 


94693 


33945 


5. 


08716 


99619 


08749 


19. 


32557 


94552 


34433 


.15 


09150 


99580 


09189 


.15 


32970 


94409 


34922 


.30 


09585 


99540 


09629 


.30 


33381 


94264 


35412 


.45 


10019 


99497 


10069 


.45 


33792 


94118 


35904 


6. 


10453 


99452 


10510 


20. 


34202 


93969 


36397 


.15 


10887 


99406 


10952 


.15 


34612 


93819 


36892 


.30 


11320 


99357 


11394 


.30 


35021 


93667 


37388 


.45 


11754 


99307 


11836 


.45 


35429 


93514 


37887 


7. 


12187 


99255 


12278 


21. 


35837 


93358 


38386 


.15 


12620 


99200 


12722 


.15 


36244 


93201 


38888 


.30 


13053 


99144 


13165 


.30 


36650 


93042 


39391 


.45 


13485 


99087 


13609 


.45 


37056 


92881 


39896 


8. 


13917 


99027 


14054 


22. 


37461 


92718 


40403 


.15 


14349 


98965 


14499 


.15 


37865 


92554 


40911 


.30 


14781 


98902 


14945 


.30 


38268 


92388 


41421 


.45 


15212 


98836 


15391 


.45 


38671 


92230 


41933 


9. 


15643- 


98769 


16838 


23. 


39073 


92050 


42447 


.15 


16074 


98700 


16286 


.15 


39474 


91879 


42983 


.30 


16505 


98629 


16734 


.30 


39875 


91706 


43481 


.45 


16935 


98556 


17183 


.45 


40275 


91531 


44001 


10. 


17365 


98481 


17633 


24. 


40674 


91355 


44522 


.15 


17794 


98404 


18083 


.15 


41072 


91176 


45046 


,30 


18224 


98325 


18534 


.30 


41469 


90996 


45572 


.45 


18652 


98245 


18986 


.45 


41866 


90814 


46100 


11. 


19081 


98163 


19438 


25. 


42262 


90631 


46630 


.15 


19509 


98079 


19891 


.15 


42657 


90446 


47163 


.30 


19937 


97992 


20345 


.30 


43051 


90259 


47697 


.45 


20364 


97905 


20800 


.45 


43445 


90070 


48234 


12. 


20791 


97815 


21256 


26. 


43837 


89879 


48773 


.15 


21218 


97723 


21712 


.15 


44229 


89687 


49314 


.30 


21644 


97630 


22169 


.30 


44620 


89493 


49858 


.45 


22070 


97534 


22628 


.45 


45010 


89298 


50404 


13. 


22495 


97437 


23087 


27. 


45399 


89101 


50952 


' .15 


22920 


97338 


23547 


.15 


45787 


88902 


51503 


.30 


23345 


97237 


24008 


.30 


46175 


88701 


52056 


.45 


23769 


97134 


24470 


.45 


46561 


88499 


52612 



126 



NATtJRAL SINES, COSIxNES, AND TANGENTS. 



Table — (Continued ). 



D.M. 


Sine. 


Cosine. 


Tangent. 


DM. 


Sine. 


Cosine. 


Tangent. 


28. 


46047 


88295 


53170 


37. 


60182 


79864 


75355 


.15 


47332 


88089 


53731 


.15 


60529 


79600 


76041 


.30 


47716 


87882 


54295 


.30 


60876 


79335 


76732 


.45 


48099 


87673 


54861 


.45 


61222 


79069 


77428 


29. 


48481 


87462 


55430 


38. 


61566 


78801 


78128 


.15 


48862 


87250 


56002 


.15 


61909 


78532 


78833 


.30 


49242 


87036 


56577 


.30 


62251 


78261 


79543 


.45 


49622 


86820 


57154 


.45 


62592 


77988 


80258 


30. 


50000 


86603 


57735 


39. 


62932 


77715 


80978 


.15 


50377 


86384 


58318 


.15 


63271 


77439 


81703 


.30 


50754 


86163 


58904 


.30 


63608 


77162 


82433 


.45 


51129 


85941 


59493 


.45 


63944 


76884 


83169 


31. 


51504 


85717 


60086 


40. 


64279 


76604 


83910 


.15 


51877 


85491 


60681 


.15 


64612 


76323 


84656 


.30 


52250 


85264 


61280 


.30 


64945 


76041 


85408 


.45 


52621 


85035 


61881 


.45 


65276 


75756 


86165 


32. 


52992 


84805 


62486 


41. 


65606 


75471 


86928 


.15 


.53361 


84573 


6.3095 


.15 


65935 


75184 


87697 


.30 


63730 


84339 


63707 


.30 


66262 


74896 


88472 


.45 


54097 


84104 


64392 


.45 


66588 


74606 


89253 


33. 


54464 


83867 


64940 


42. 


66913 


74314 


90040 


.15 


54829 


83629 


65562 


.15 


67237 


74022 


90833 


.30 


55194 


83389 


66188 


.30 


67559 


73728 


91633 


.45 


55557 


83147 


66817 


.45 


67880 


73432 


92439 


34. 


55919 


82904 


67450 


43. 


68200 


73135 


93251 


.15 


56280 


82659 


68087 


.15 


68518 


72837 


94070 


.30 


56641 


82413 


68728 


.30 


68835 


72537 


94896 


.45 


57000 


82165 


69372 


.45 


69151 


72236 


95729 


35. 


57358 


81915 


70020 


44. 


69466 


71934 


96568 


.15 


57715 


81664 


70673 


.15 


69779 


71630 


97415 


.30 


58070 


81412 


71329 


.30 


70091 


71325 


98269 


.45 


58425 


81157 


71989 


.45 


70401 


71019 


99131 


36. 


58779 


80902 


72654 


45. 


70710 


70710 


1.00000 


.15 


59131 


80644 


73323 


.15 


71019 


70401 


1.00876 


.30 


59482 


80386 


73996 


.30 


71325 


70091 


1.01760 


.45 


59832 


80125 


74673 

















TANGENTS FROM 450 TO 9(P. 






D. 


Tangent. 


D. 


Tangent. 


D. 


Tangent. 


D. 


Tangent. 


D. 


Tangent 


46 


1.0355 


55 


1.4281 


64 


2.0503 


73 


3.2708 


82 


7.1153 


47 


1.0724 


56 


1.4826 


65 


2.1445 


74 


3.4874 


83 


8.1443 


48 


1.1106 


57 


1.5399 


66 


2.2460 


75 


3.7321 


84 


9.5144 


49 


1.1504 


58 


1.6003 


67 


2.3558 


76 


4.0107 


85 


11.4301 


50 


1.1918 


59 


1.6643 


68 


2.4751 


77 


4.3314 


86 


14.3007 


51 


1.2349 


60 


1.7321 


69 


2.6051 


78 


4.7046 


87 


19.0811 


52 


1.2799 


61 


1.8040 


70 


2.7475 


79 


5.1445 


88 


28.6363 


53 


1.3270 


62 


1.8807 


71 


2.9042 


80 


5.6712 


89 


57.2900 


54 


1.3764 


63 


1.9626 


72 


3.0776 


81 


6.3137 


90 


Infinite. 



SINES AND SECANTS OF ANGLES. 127 

To find the Sine or Cosine of any Angle exceeding 45°, hy the 
foregoing Table. 

Subtract the angle given from 90, look in the table for the re- 
mainder, and opposite to it take out the sine for the cosine, and the 
cosine for the sine of the angle given. 

Example.— What is the sine and the cosine of 85°1 

85°— 90° — 5°, and opposite to 5° in the table is 08716 and 99619, 
which are respectively the cosine and sine of 85° 

The sine of 90° is 100000, cosine 0. 

The sine of an arc, divided by the cosine, gives the natural tan- 
gent of that arc. 

To compute Tangents and Secants, 

Cos. : sin. : : rad. : tangent. 

Cos. : rad. : : rad. : secant. 

Sin. : COS. : : rad. : cotangent. 

Sin. : rad. ; : rad. : cosecant. 

To find the Secant of an Angle, 
Divide 1 by the cosine of that angle. 
Example.— The cosine of 21° 30' is .93041 ; 

= 1.07479. 



.93041 



To find the Cosecant of an Angle. 
Divide 1 by the sine of the angle. 
Example. — The sine of 21° 30' is .36650; 



= 2.72951. 



.36650 

To find the Versed Sine. 
Subtract the cosine from 1. 
Example. — The cosine of 21° 30' is .93042 ; 
1— .93042 = .06958. 

To find the Cover sed Sine. 
Subtract the sine of the angle from 1. 
Example.— The sine of 21° 30' is .36650 ; 
1— .36650 = .6335. 

To find the Chord of any Angle, 
Take the sine of half the angle and double it. 
Example.— The chord of 21° 30' is required. 

^. ^21° 30' 

Sine of — ^ = .18653X2 = .37304. 



128 MECHANICAL POWERS. 



MECHANICAL POWERS. 

Power is a compound of weight, or the expansion of a body, mul- 
tiplied by its velocity : it cannot be increased by mechanical means. 

The Science of Mechanics is based upon Weight and Power, or 
Force and Resistance. 

The weight is the resistance to be overcome, the power is the 
requisite force to overcome that resistance. When they are equal 
no motion can take place. 

The Powers are three in number, viz., Lever, Inclined Plane, 
and Pulley. 

Note.— The Wheel and Axle is a continual or revolving lever, the Wedge is a 
double inclined plane, and the Screw is a revolving inclined plane. 



LEVER. 

When the Fulcrum (or Support) of the Lever is between the Weight and 
the Power. 

Rule.— Divide the weight to be raised by the power, and the 
quotient is the difference of leverage, or the distance from the ful- 
crum at which the power supports the weight. 

Or, multiply the weight by its distance from the fulcrum, and the 
power by its distance from the same point, and the weight and 
power will be to each other as their products. 

Example.— A weight of 1600 lbs. is to be raised by a force of 80 
lbs. ; required the length of the longest arm of the lever, the short- 
est being 1 foot. 

15^>ll = 20feet,4n.. 
80 

Proof, by second rule. 

1600 X 1=1600. 
80X20 = 1600. 
Example.— A weight of 2460 lbs. is to be raised with a lever 7 
feet long and 300 lbs. ; at what part of the lever must the fulcrum 
be placed 1 

51— = 8.2 ; that is, the weight is to the power as 8.2 to 1 ; there- 
300 ' 

7 V 12 84 
fore the whole length -— — - = — == 9.13 inches, the distance of 
o.-4~rl u.Z 

the fulcrum from the weight. 

Example.— A weight of 400 lbs. is placed 15 inches from the ful- 
crum of a lever ; what force will raise it, the length of the other 
arm being 10 feet 1 

400X15 _,, . 

= 50 lbs., Ans. 

120 

NoTK.— Pressure upon fulcrum equal the sum of weight and power. 



MECHANICAL POWERS. 129 

When tke fulcrum is at one Extremity of the Lever, and the Poioer or 
the Weight, at the other. . ' 

^ Rule.— As the distance between the power or weight and ful- 
crum is to the distance between the w^eight or power and fulcrum 
80 is the effect to the power. * 

Example.— What power will raise 1500 lbs., the weight beinff 5 
feet from it, and 2 feet from the fulcrum 1 

5+2 = 7 : 2 : : 1500 : 428.5714+^715. 
Example.— What is the weight on each support of a beam that is 
30 feet long, supported at both ends, and bearing a weight of 6000 
lbs. 10 feet from one end 1 

30 : 20 : : 6000 : 4000 lbs. at the end nearest the weight ; and 
30 : 10 : : 6000 : 2000 lbs. at the end farthest from th'e weight. 
Note.— Pressure upon fulcrum is the difference of the weight and the power. 
The General Rule, therefore, for ascertaining the relation of 
Power to Weight m a lever, whether it be straight or curved, is 
the power multiplied by its distance from the fulcrum, is equal to 
the weight multiplied by its distance from the fulcrum. 

Let P be called the power, W the weight, p the distance of P 
from the fulcrum, and w the distance of W from the fulcrum ; then 

P : W : :-w; : p, orPx;7 = Wxi^: 
and 

V w 

'Wxw_ Vxv 

If several weights or powers act upon one or both ends of the 
lever, the condition of equilibrium is 

PXp+FX/+P^^X;?^ &c.,=z.WXl^;+W'x^^;^ &c. 
In a system of levers, either of similar, compound, or mixed kinds, 
the condition is * 

Px^x/x/;^^^ 

wXw'Xw" 

.Jt\^f"'^ lb-^,f^d./ each 10 feet,/' 1 foot ; and if tt- and z.,' be 
each 1 foot, and w'' 1 inch, then 

1 X 120 X 120 X 12 __ 172800 

12x12x1 — 144 — 1^^^ 5 ^hat is, 1 lb. will balance 

1200 lbs. with levers of the lengths above given. 

rpJIt^Jnf";?'^'^^ ^-^^^ ""^ ^^^ l^""^'^ ^^ ^^® ^^^^^ formula are not considered, the 
centre of gravity being assumed to be over the fulcrums. 

If the arms of the lever be equally bent or curved, the distances 
from the fulcrum must be measured upon perpendiculars, drawn 
from the lines of direction of the weight and power, to a line run- 
nmg horizontally through the fulcrum ; and if unequally curved 
measure the distances from the fulcrum upon a line running hori- 
zontally through it till it meets perpendiculars falling from the ends 
of the lever. 



130 MECHANICAL POWERS. 



WHEEL AND AXLE. 

The power multiplied by the radius of the wheel is equal to the 
weight multiplied by the radius of the axle. 

As the radms of the wheel is to the radius of the axle, so is the 
effect to the power. ^ ^^ . . , ^ 

When a series of wheels and axles act upon each other, either by 
belts or teeth, the weight or velocity will be to the power or unity 
as the product of the radii, or circumferences of the wheel-s, to the 
product of the radii, or circumferences of the axles. 

Example.— If the radii of a series of wheels are 9, 6, 9, 10, and 
12, and their pinions have each a radius of 6 inches, and the weight 
applied be 10 lbs., what weight will it raise] 

6X6X6X6X6 

Or, if the 1st wheel make 10 revolutions, the last will make 75 in 
the same time. 

Note.— For a fuller treatise on wheels, see Grier's Mechanic's Calculator, pages 
130 to 136. 

To find the Power of Cranes, <$fc. 

Rule.— Divide the product of the driven teeth by the product of 
the drivers, and the quotient is the relative velocity, which, multi- 
plied by the length of the winch and the force in lbs., and divided 
by the radius of°the barrel, will give the weight that can be raised. 

Example.— A force of 18 lbs. is applied to the winch of a crane, 
the length being 8 inches ; the pinion having 6, the wheel 72 teeth, 
and the barrel 6 inches diameter. 



-^ = 12X8X18 — 1728-^-3 = 576 lbs. weight. 

Let w represent length of winch, ^^^'P 

r ** radius of barrel, 



W. 
r 
Wr = vwV. 

vw 



p " force applied 

7, " velocity, 

W " weight.raised. 
Example —A weight of 94 tons is to be raised 360 feet in 15 
minutes, by a force the velocity of which is 220 feet per minute ; 
what is the power required ] 

OCA 

= 24 feet per minute. 

15 

?i^= 10.2542 tons. 
220 

In a wheel and axle, where the axle has two diameters, the con- 
dition of equilibrium is 

W : P : : R : 4 [r—r') ; 
or, PxR==Wxi(r— r^j; .^^ 

that is, the weight is to the power as the lever by which the power 
works, is to half the difference of the radii of the axle ; 



MECHANICAL POWERS. 131 

R representing radius of wheel, 
r " radius of large axle, 

r' " radius of less axle. 



INCLINED PLANE. 



Rule.— As the length of the plane is to its height, so is the 
weight to the power. 

Example.— Required the power necessary to raise 1000 lbs. up 
an inclined plane 6 feet long and 4 feet high. 

As 6 : 4 : : 1000 : 666.66 Ans. 

Wxh 



Let W represent weight, 



height of plane, 

length of plane, 

power, 

base of plane, 

pressure on plane. 



h 
Wxb 

-r-=P' 



To find the Length of the Base, Height, or Length of the Plane, 
when any two of them are given. 

Rule. — For the length of the base, subtract the square of the 
height from the square of the length of the plane, and the square 
root of the remainder will be the length of the base. 

For the length of the plane, add the squares of the two other di- 
mensions together, and the square root of their sum will be the 
length required. 

For the height, subtract the square of the base from the square of 
the length of the plane, and the square root of the remainder is the 
height required. 

Example. — The height of an inclined plane is 20 feet, and its 
length 100 ; what is its base, and the pressure of 1000 lbs. upon the 
plane "? 

-v/202— 100^ = 9600 = 97.98 the base. 

As 100 : 20 : : 1000 : 200 lbs. necessary power to raise the 1000 
^ 1000X97.98 
lbs., and — = 979.8 the pressure upon the plane. 

If two bodies on two inclined planes sustain each other hy the aid of 
a cord over a pulley, their weights are directly as the lengths of the 
planes. 

Example.— If a body of 50 lbs. weight, upon an inchned plane, of 
10 feet rise in 100, be sustained by another weight on an opposite 
plane, of 10 feet rise to 90 of an inclination, what is the weight of 
the latter 1 

As 100 : 90 : : 50 : 45, the answer. 

When a body is supported by two planes, and if the weight be repre- 
sented by the sine of the angle between the two planes. The pressures 
upon them are reciprocally as the sines of the inclinations of those 
planes to the horizon, viz. : 



132 MECHANICAL POWERS. 

( Sine of the angle be- 

_, . , ^ ^1 tween the planes. 

The weight, ) , g^^g ^f t^g l3 ^^ 

The pressure upon one plane, > are as<^ ^^^ 1^^^ ^ 

The pressure upon the other plane, ) j g.^^ ^^ ^^^ ^^^^^ ^^ 

\ the other plane. 

Thus, if the angle between the planes was 90°, of one plane 60"^, 
and the other 30°— since the natural sines of 90°, 60°, and 30° are 
1, .866, and .500— if the body weighed 100 lbs., the pressure upon 
the plane of 30° would be 86.6 lbs., and upon the plane of 60°, 50 
lbs. = the centre of gravity being in the centre of the body. 

When the 'power does not act parallel to the plane, draw a line per- 
pendicular to the direction of the power's action from the end of 
the base line (at the back of the plane), and the intersection of this 
line on the length will determine the length and height of the plane. 
Note.— When the line of direction of the power is parallel to the plane, the 
power is least. 

The space which a body describes upon an inclined plane, when 
descending on the plane by the force of gravity, is to the space it 
would freely fall in the same time, as the height is to the length of 
the plane ; and the spaces being the same, the times will be in- 
versely in this proportion. 

Example. — If a body be placed upon an inclined plane 300 feet 
long and 25 feet high, what space will it roll down in one second 
by the force of gravity alone 1 

As 300 : 25 : : *16.08 : 1.33 feet, Ans. 

If a body be projected down an inclined plane with a given velocity, 
then the distance which the body will be from the point of projec- 
tion in a given time will be tXv-\—Xl6.08t^ ; but if the body be 



* The distance a body will freely fall in one second by the force of gravity. 

The force of an inclined plane bears the same proportion to the force of gravity 
as the height of the plane bears to its length ; that is, the force which accelerates 
a body down an inclined plane is that fractional part of the force of gravity which 
is represented by the height of the plane divided by its length. 

Let h represent the height of the plane, I its length, t the time in seconds, s the 
space which a body will move through in a given time, v the velocity, and i the 

/ h\ 

angle of inclination \^sm. ^ = jj* 

16.08 hf^ tv Zr^ v^ ^ ^„ , 

- or ^' or -r— , or ^JT^^-;' ^r sin. zXl6.08«2. 



tj:=z — ,or — -, — ,or V — - — , or sin. 2X32.16t, or ^sm. zXo4.3s 
« = -' or ^5-^^, or V -— -, or ^oir «i„ ,' o^ V T 



32.16 h' ^ 16.08 li 32.16 sin. t' ^ 16.08 sin. i 

l °' '*^- *■ = 32i6-t' "' 16:^7^' °' ^b^- 

The acceleratmg force on the plane is to the accelerating force of gravity as v^ 
is to 64.3X5. 

If sin. i = i, it shows that the length of the plane is twice its height, or ^ = 30©. 

If the proportion wliich the length of the plane bears to the height be given, sub- 
stitute these proportions for the length and height in the above rules, and the con- 
clusions will be equally true. 



MECHANICAL POWERS. 133 

projected upward, then the distance of the body from the point of 
projection will he tXv—jXl^Mt^, 



WEDGE. 

WIi£n two Bodies are forced from mie another, in a direction Parallel to 
the back of the Wedge. 

Rule.— As the length of the wedge is to half its back, so is the 
resistance to the force. 

Example.— The length of the back of a double wedge is 6 inches 
and the length of it through the middle 10 inches ; what is the pow- 
er necessary to separate a substance having a resistance of 150 lbs. 1 
As 10 : 3 : : 150 : 45 lbs., Ans. 

When only one of the Bodies is Movable. 

Rule.— As the length of the wedge is to its back, so is the resist- 
ance to the power. 

Example.— What power, applied to the back of a wedge, will raise 
a weight of 15,000 lbs., the wedge being 6 inches deep, and 100 Ion? 
on its base. ^ 

As 100 : 6 : : 15000 : 900 lbs., Ans. 

Note.— As the power of the wedge in practice depends upon the split or rift in 
the vyod to be cleft, or in the body to be raised, the above rules are only theo- 
reucal where a rift exists. 



SCEEW. 

As the screw is an inclined plane wound round a cylinder, the 
length of the plane is found by adding the square of the circumfer- 
ence of the screw to the square of the distance between the threads, 
and taking the square root of the sum, and the height is the distance 
between the consecutive threads. 

Rule.— As the length of the inclined plane is to the pitch or 
height of It, so IS the weight to the power. 

When a wheel or capstan is applied to turn the screw, the length 
of the lever IS the radius of the circle described by the handle of the 
wheel or capstan bar. 

Let P represent power. 



R 

W 

/ 

P 

X 



length of lever, 
weight, 

length of the inclined plane, 
pitch of screw or height of plane, 
effect of power at circumference of screw, 
radius of screw. 
M 



134 



MECHANICAL POWERS. 



w 


:P, 


p 


:P, 


p 


• V^ 


p 


w, 


V 


: U 


p 


. X, 


r 


■■ R, 


X 


: P. 



Then, by the above rules, 

As Z : j!7 
I :W 
W; / 
p: I 
P :W 
r : R 
P: X 
R: r 

Example.— What is the power requisite to raise a weight of 8000 
lbs. by a screw of 12 inches circumference and 1 inch pitch ^ 
122+12 rr: 145, and ^145 =: 12.04159. 
Then, 12.0416 : : 1 : : 8000 : 664.36 lbs., Ans. 
And if a lever of 30 inches length was added to the screw, 
12—3.1416 = 3.819+2+30 = 31.9095, length of lever. » 
Then, as 31.9095 : 1.9095 : : 664.363 : 39.756 lbs., Ans. 
Or, when the circumference described by the power is used (C), 
we have 

P : W: : ;? : C, 

C : ;> : : W : P, 

PXC=WX;?; 

thus, 39.756 : 8000 : : 1 : 201.227 = circumference described by 

lever, which is the hypothenuse of the triangle formed by the base 

and height of the inclined plane. 

When a hollow screw revolves upon one of less diameter and 
pitch (as designed by Mr. Hunter), the effect is the same as tb^t of 
a single screw, in which the distance between the threads is equal 
to the difference of the distances between the threads of the two 
screws. 

If one screw has 20 threads in an inch pitch, and the other 21, the 
power is to the weight as the difference between ^V ^^^ Tf* ^^ tIo 
— 1 to 420. 

In a complex machine, composed of the screw, and wheel, and 
axle, the relation between the weight and power is thus : 
Let X represent the effect of the power on the wheel, 
R " the radius of the wheel, 
p " the pitch of the screw, 
r " the radius of the axle, 
C " the circumference described by the power. 
Then, by the properties of the screw, 
PxC=a:X;?; 
and of the wheel and axle, 

a;XR==WXr. 

Hence we have 

PxCX2:XR = xX:pXWXr. 

Omitting the common multiplier, x, 

PXCXR — WX;?Xr; 
or P: W: -.pXr: CxR, 
andj^Xr :CXR : : P : W. • 



MECHANICAL POWERS. 135 

Ex AMPLE. —■What weight can be raised with a power of 10 lbs. 
applied to a crank 32 inches long, turning an endless screw of 3^ 
inches diameter and one inch pitch, applied to a wheel and axle of 
20 and 5 inches in diameter respectively ] 
Circumference of 64 = 201. 

1 : 201 : : 10 : 2010. 
Radii of wheel and axle, 10 and 2.5. 

2.5 : 10 : : 2010 : 8040 lbs., Arts., 
or 2.5X1 : 201X10:: 10: 8040. 

And when a series of wheels and axles act upon each other, the 
weight will be to the power as the continued product of the radii of 
the wheels to the continued product of the radii of the axles ; 
thus, W : P : : R3 : r^ ; 
or, r^ : R3 : : P : W, 
there being three wheels and axles of the same proportion to each 
other. 

Example. — If an endless screw, with a pitch of half an inch, and 
a handle of 20 inches radius, be turned with a power of 150 lbs., 
and geared to a toothed wheel, the pinion of which turns another 
wheel, and the pinion of the second wheel turns a third wheel, to 
the pinion or barrel of whi^ih is hung a weight, it is required to 
know what weight can be sustained in that position, the diameter 
of the wheels being 18, and the pinions 2 inches '! 
pXr' : CxR^ :P: W; 
or .5X1^ : 125.6X93 : : 150; 
which, when extended, gives 

.5 : 91562.4 : : 150 : 27468720 lbs., Ans. 



PULLEY. 

When only one Cord or Rope is used. 

Rule. — Divide the weight to be raised by the number of parts of 
the rope engaged in supporting the lever or movable block. 

* Example. — What power is required to raise 600 lbs. when the 
lower block contains six sheaves and the end of the rope is fasten- 
ed to the upper block, and what power when fastened to the lower 
block] 

_ = 50 lbs., 1st Ans. 

6X2 

^^^ ■ = 46.15 lbs., 2d ^r?5.; 



6x2+1 

or W = nXP, 
n signifying the number of parts of the rope which sustain the 
lower block. 



136 MECHANICAL POWERS. 

Wken TTwre than one Rope is used. 

In a Spanish burton, where there are two ropes, tw^o moveable pul- 
leys, and one fixed and one stationary pulley, with the ends of one 
rope fastened to the support and upper moveable pulley, and the ends 
of the other fastened to the lower block and the power^ the weight 
is to the power as 5 to 1. 

And in one where the ends of one rope are fastened to the sup- 
port and the power, and the ends of the other to the lower and up- 
per blocks, the weight is to the power as 4 to 1. 

I?i a system of pulleys, with any number of ropes, the ends being fas- 
tened to the support, 

W = 2^XP, 
n expressing the number of ropes. 

Example.— What weight will a power of 1 lb. sustain in a system 
of 4 movable pulleys and 4 ropes'? 

1X2x2x2x3 — 16 \\)S,Ans, 
When fixed pulleys are used in the place of hooks, to attach the ends 
of the rope to the support, 

W = 3«XP. 
Example.— What weight will a powder of 5 lbs. sustain with 4 
moveable and 4 fixed pulleys, and 4 ropes 1 

5x3x3x3x3 = 405 lbs., Ans. 

When the ends of the rope, or the fixed pulleys, are fastened to the 
weight, 

W=r(2"— 1)XP, 

andW==(3^— 1)XP, 

which would give, in the above examples, 

1X2X2X2X2= 16—1=: 15 lbs., 
5X3X3X3X3 = 405— 1=404 lbs. 



CENTRES OF GRAVITY. 137 



CENTRES OF GRAVITY. 

The Centre of Gravity of a body, or any system of bodies con- 
nected together, is the point about which, if suspended, all the parts 
are in equilibrio. 

If the centres of gravities of two bodies B C be connected by a 
line, the distances of B and C from the common centre of gravity 
A will be as the weights of the bodies ; 

thus, B : C : : CA : AB. 

SURFACES. 
1. To find the Centre of Gravity of a Circular Arc. 

Radius of circle X chord of arc ,.■ ^ , 

• — — — — — = distance from the centre of the 

length of the arc 

circle. 

2. Of a Parallelogram^ Rhombus, Rhomboid, Circle, Ellipse, Regular 
Polygon, or Lmne. 
The geometrical centre of these figures is their centre of gravity. 

3. Of a Triangle. 
On a line drawn from any angle to the middle of the opposite side, 
at § of the distance from the angle. 

4:. Of a Trapezium. 
Draw the two diagonals, and find the centres of gravity of each 
of the four triangles thus formed ; join each opposite pair of these 
centres, and the intersection of the two lines will be the centre of 
gravity of the figure. 

5. Of a Trapezoid. 
On a line a, joining the middle points of the two parallel sides 

B h, the distance from B = ^x(~—r\, 
3 ^ B+o / 

Q. Of a Sector of a Circle. 

2 X chord of arc X radius of circle ... 

5-— -j -r--^ = distance from the centre of 

3 X length of arc 

the circle. 

1. Of a Semicircle. 

4 X radius of circle 

= distance from centre. 

oXo.i4lD 

8. Of a Segment of a Circle. 

Chord of the segment 3 ,. . , 

-rr-- 7 — = distance from the centre. 

12 X area of segment 

9. Of a Parabola. 

Distance from the vertex, § of the abscissa. 

M2 



138 CENTRES OF GRAVITY. 

10. Of any Plane Figure. 
Divide it into triangles, and find the centre of gravity of each ; 
connect two centres together, and find their common centre ; then 
connect this and the centre of a third, and find the common centre 
of these, and so on, always connecting the last found common cen- 
tre to another centre till the whole are included, and the last com- 
mon centre will be that which is required. 

11. Of a Cylinder^ Cone^ Fnistiim of a Cone^ Pyramid^ Frustum of a 
Pyramid, or Ungula. 
The centre of gravity is at the same distance from the base as 
that of the parallelogram, triangle, or trapezoid, which is a right 
section of either of the above figures. 

12. Of a Sp/i£re, Spherical Segment, or ZoTie, 
At the middle of their height. 

SOLIDS, 

I. of a Sphere, Cylinder, Cube, Regular Polygon, Spheroid^ Ellipsoid^ 
Cylindrical Ring, or any Spindle. 
The geometrical centre of these figures is their centre of gravity. 

2. Of a Right Ungula, Prism, or Wedge. 
At the middle of the line joining the centres of the two ends. 

3. Of a Prismoid, or Ungula. 
At the same distance from the base as that of the trapezoid or 
triangle, which is a right section of them. 

4. Of a Pyramid, or Cone. 
Distance from the base, i of the line joining the vertex and cen- 
tre of gravity of the base. 

b.Ofa Frustum of a Cone, or Pyramid. 
Distance from the centre of the smaller end, 

= i height x ^^^J^.^^^ ; 

R and r radii of the greater and less ends in a cone, and the sides 
of a pyramid. 

6. Of a Parabolaid. 
Distance from the vertex, f of the abscissa. 

1. Of a Friistum of a Paraboloid. 
Distance on the abscissa from the centre of the less end, 

i h -TjTT-T' ^ being the height. 

S.Ofa Spherical SegmeTit, 
Distance from the centre, 

^ ^"T"/ , V being the versed sine, s the solid contents of the 
s 

segment, and r the radius of the sphere. 



GRAVITATION. 
9. Of a Spherical Sector. 



139 



Distance from the centre, | (r — ). 

10. Of any System of Bodies, 
Distance from a given plane, 
♦ BD+B'D'+B-D-+, &c. ^ , . ^^ ,., ^ ^ 
= Yx — p/ , p// ^ > -^ bemg the solid contents or weights, 

and D the distances of their respective centres of gravity from the 
given plane. 



GRAVITATION. 



In bodies descending freely by their ovfn weight, the velocities 
are as the times, and the spaces as the square of the times. 
The times, then, will be 1, 2, 3, 4, &c. ; 
The velocities, then, will be 1, 2, 3, 4, &c. ; 
The spaces passed through as 1, 4, 9, 16, &c. ; 
And the spaces for each time as 1, 3, 5, 7, 9, &c. 
A body faUing freely will descend through 16.0833 feet m the first 
second of time, and will then have acquired a velocity which will 
carry it through 32.166 feet in the next second. 



Table exhibiting the Relation of Time^ Space, and Velocities. 



Seconds from 
the begin- 
nia^ of the 


Velocity acquired at 
the end of that time. 


Squares. 


Space fallen through 
in that tiiuc. 


Spaces. 


Space fallen 

through in the 

last Second oi the 


descent. 










fall. 


1 


32.166 


1 


16.08 


1 


16.08 


2 


64.333 


4 


64.33 


3 


48.25 


3 


96.5 


9 


144.75 


5 


80.41 


4 


128.665 


16 


257.33 


7 


112.58 


5 


160,832 


25 


402.08 


9 


144.75 


6 


193. 


36 


579. 


11 


176.91 


7 


225.166 


49 


788.08 


13 


209.08 


8 


257.333 


64 


1029.33 


15 


241.25 


9 


289.5 


81 


1302.75 


17 


273.42 


10 


321.666 


100 


1608.. 33 


19 


305.58 


11 


353.832 


121 


1946.08 


21 


337.75 


12 


386. 


144 


2316. 


23 


369.92 



and in the same manner the table might be continued to any extent. 

To find the Velocity a Falling Body will acquire in any Given 

Time, 

RuLE.—Multiply the time in seconds by 32.166, and it will give 
the velocity acquired in feet per second. 
Example^ — Required the velocity in 12 seconds. 
12X32.166 = 386 feet, ^n*. 



140 GRAVITATION. 

To find the Time which a Body will he in falling through a Given 

Space, 
Rule. — Divide the space in feet by 16.083, and the square root of 
the quotient will give the required time in seconds. 

Example. — How long will a body be in falling through 402.08 feet 
of space I 

^402.08-M6.083 = 5, Ans. 

To find the Space through which a Body will fall in any Given 

Time, 
Rule. — Multiply the square of the time in seconds by 16.083, and 
it will give the space in feet. 

Example. — Required the space fallen through in 5 seconds. 
52 =: 25X16.083 =402.08 feet, Ans. 

To find the Velocity a Body will acquire by falling from any 

Given Height. 
Rule. — Multiply the space in feet by 64.333, and the square root 
of the product will be the velocity acquired in feet per second. 

Example. — Required the velocity a ball has acquired in descend- 
ing through 579 feet. 

^579x64.333== 193 feet, Ans. 
Or, when the time is given, multiply the time in seconds by 32.166, 
Thus, the time for 679 feet is 6 seconds; then, 6x32.166 = 
192.996, Ans. 

To find the Space fallen through, the Velocity being given. 
Rule. — Divide the velocity by 8, and the square of the quotient 
will be the distance fallen through to acquire that velocity. 

Example. — If the velocity of a common ball is 579 feet per see-- 
ond, from what height must a body fall to acquire the same velocity T 
579-^8 = 72.3752 = 5237 feet, Ans. 

To find the Time, the Velocity per Second being given. 
Rule. — Divide the given velocity by 8, and i of the quotient is 
the answer. 

Example. — How long must a bullet be falling to acquire a velo- 
city of 800 feet per second 1 

800-^-8 = 100-^4 = 25 seconds, Ans. 
Let s represent the space described by any falling body, t the time, 
and V the velocity acquired in feet. 

tv v^ 

Then 5 = 16.08 i% or-, or -— . 
Z d4.o 

, 5 V 2s 

2* 
V = 2-/16.08 5, or 32.16 t, or — . 

The distance fallen through iyi feet is very nearly equal to the square 
of the time in fourths of a second. 



GRAVITATION. 141 

Example.— A bullet being dropped from the spire of a church, was 
4 seconds in reaching the ground ; what was the height ^ 
4x4xi6zrz256feet, Ans. 

Example.— What is the depth of a well, a buUet being 2 seconds 
m reaching the bottom 1 ^ 

2X4X8 = 64 feet, ^n5. 
Or, more correctly, as in case 2, 

4X4X16.0833 = 257.33 feet, 
and 2X2X16.0833= 64.33 feet. 
Bt/ Inversion. 
In what time will a bullet fall through 256 feet 1 

^256 = 16, and 16—4 = 4 seconds, Ans. 
Ascending bodies are retarded in the same ratio that descending bodies 
are accelerated. ^ 

To find the Space moved through by a Body projected upward or 
downward with a Given Velocity. 
If projected dovmward. 
Rule.— Multiply the square of the time in seconds by 16 083 the 
velocity of the projection in feet by the number of seconds the body 
IS m motion, and the sum of these products is the answer. 
If projected upward. 
Then the difference of the above products wiU give the distance 
of the body from the point of projection 
Or, ^Xu±16.083x/^ 

Example.— If a shot discharged from a gun return to the earth in 
12 seconds, how high did it ascend 1 
The shot is half the time in ascending. 

12-^2 =. 6, and 6 ^ x 16.083 = 579 feet, Ans. 
Or, 62x16.083—192.96x6. 

Example.— If a body be projected upward with a velocity of 30 
feet per second, through what space will it ascend before it beffins 
to return 1 ^ 

302^64.3= 13.9 feet, ^715. 
Example.— If a body be projected upward with a velocity of 100 
feet per second, it is required to find the place of the body at the 
end of 10 seconds. 

\^iio ^^f ^^^^' ^^^ ^P^^^ if gravity did not act, and 16.083xi0« 
= 1608.3, the loss arising from gravity. 
Hence 1000-1608.3 = 608.3 feet below the point of projection. 

To find the Velocity of a Falling Stream of Water per Second 
{the perpendicular distance being given) at the End of any 
Given Time. *^ ^ 

r.^T"'"^^ two bodies begin to descend from rest, and from the same point, the 

Wh? ^/"''^'r'^ P'"^"^' ^"?/?^ °^^^^ ^^"i"S freely, their velocities at all equal 
heights below the surface will be equal. ° •' » «" cquai 

The space through which a body will descend on an inclined plane is to the 

^ane to uTfengTh.' '^ ^^^^ ^''^^^ ^ '^^ '^"'^ "^""^ ^ the height of thi 

If a body descend in a curve, it suffers no loss of velocity. 



142 GRAVITIES OF BODIES. 

Example. — One end of a sluice is 30 inches lower than the other, 
what IS the velocity of the stream per second] 

By case 4, 30 inches = 2.5 feet, x 64.33== 160.82, and ^^160.82 
= 12.65 feet, Ans. 

What is the distance a stream of water will descend on an incli- 
ned plane 10 feet high, and 100 feet long at the base, in 5 seconds'? 

5- X 16.083 ==402.08, and 100 : 10 • : 402.08 : 40.20 feet, Ans. 

The momenlum with which a falling body strikes is equal to its weight 
multiplied by its velocity. 

If a weight of 4500 lbs. fall through 10 feet, with what force does 
it strike 1 

yi0x64.33r=25.35x4500==114075Ibs.,^7i5. 

If a stream of salt water, running at the rate of 5 feet per second, 
strike a dam 15 by 4 feet, what is the pressure of the stream] 

Rule. — Multiply the height of the fall by the weight of the fluid, 
and the product by the area of the resisting body, and that product, 
again, by the velocity in feet per second. 

By case 5, 5-^8=^.625-==.390625, the height of the fall of the 
water, x64 lbs., the weight per cubic foot, =25 lbs. X(15x4)60 
=1500x5=7500 pounds, Ans. 

Note. — Water being a yielding- substance, an allowance for loss of power should 
be made. 



PROMISCUOUS EXAMPLES. 



1. Suppose a bullet to be 1 minute in falling, how far will it fall 
in the last second ] 

Space fallen through equal the square of the time ; then 1 minute 
=60 seconds. 

60^ X 16.083=57898 distance for 60 seconds, 
59^X16.083 =55984 " " 59 

1914 *' " 1 " Ans. 

2. Find the time of generating a velocity of 193 feet per second, 
and the whole space descended. 

193-^32.166= 6 seconds, ) . 
62 X 16.083=579 feet, S 

The velocity acquired at any period is equal to twice the mean velocity 
during that period. 

3. Then, if a ball fall through 2316 feet in 12 secoiMs, with w^hat 
velocity will it strike 1 

2316—12=193x2 = 386 feet, ^n5. 



SPECIFIC GRAVITIES. 143 

GRAVITIES OF BODIES. 

The gravity of a body, or its weight above the earth's surface 
decreases as the square of its distance from the earth's centre in 
semi-diameters of the earth. 

Example.— If a body weigh 900 lbs. at the surface of the earth 
what will It weigh 2000 miles above the surface ? ' 

The earth's semi-diameter is 3993 miles (say 4000) 

Then 2000+4000=6000 or H semi-diameters 
and 900-^ 1.5^==: 400 lbs, Ans. 
Inversely, if a body weigh 400 lbs. at 2000 miles from the earth's 
surface, what will it weigh at the surface ? 

400X1.5^=900 lbs, Ans. 
EXAMPLE.--.A body at the earth's surface weighs 360 lbs. : how 
high must i t be elevated to weigh 40 lbs. 1 
v/360-^40=3, or 3 semi-diameters from the earth's surface, Ans 

^/4000^3^=5656— 4000=1656 miles, Ans. 

fh.V.thfTT^ '^ ^r.^^'^'' ^' ^^"«^ ^^^ their^ densities different, 
the weight of a body on their surfaces will be as their densities. 

/a/ ^^'ir f'""^'^]''. ^^.^?^^^ ^nd their diameters different, the weight of 
them will be as their diameters. ujci^ul oj 

the{r%duc7s^''' "'''^ ^'''''^''' """' ^'^^ ^'-^'"'''^^ ^^' ^^i"^^ ^^'^^ ^e as 
wh^at wilM.T^^ a body weigh 10 lbs. at the surface of the earth, 

^92 anT 00 Tlf^^' 'r^ '^'^^"^ ^^ '^^ '"^ • '^^'' ^^"^ities being 
^^'i and 1 00, and their diam eters 8000 and 883000 miles. 

883000X100X10-^8000^^392=281.5 lbs., Ans. 



SPECIFIC GRAVITIES. 

wJr. ^f ^^fi^^^^i^ of a body is the proportion it bears to the 
weight of another body of known density, and water is well adapt- 
ed for the standard ; and as a cubic foot of it weighs 1000 ounces 
avoirdupois, its weight is taken as the unit, viz., 1000. 

Tojind the Specific Gravity of a Body heavier than Water. 

en^J'^^thTn^f ^^1.'^ ^""^'^ ^"^. "^^^ ^^ ^^^^^' ^"d take the differ- 
f. ?ooo tn ?h. t^^./^^gh^lost in water is to the whole weight, sa. 
is 1000 to the specific gravity of the body. 

i^'^h«^''hnf""~^^^^ ''.^^^ 'P^^^^^ ^^^^^*y of a stone which weighs 
15 lbs., but in water only 10 lbs. \ ^ 

15—10=5."' 5 : 15 : : 1000 : 3000, Ans. 
When the Body is lighter than Water. 

..^^A^l^^''^^ ^"^ ^^ ^ P^^^^ Of metal or stone, weigh the piece 
added and the compound mass separately, both in and out of watej^ 



144 SPECIFIC GRAVITIES. 

find how much each loses in water by subtracting its weight in 
water from its weight in air, and subtract the less of these differ- 
ences from the greater ; then, 

As the last remainder is to the weight of the light body in air, so 
is 1000 to the specific gravity of the body. 

Example. — What is the specific gravity of a piece of wood that 
weighs 20 lbs. in air ; annexed to it is a piece of metal that weighs 
24 lbs. in air and 21 lbs. in water, and the two pieces in water 
weigh 8 lbs. 1 

20+24—8=36 
24—21 z=3^ 

33": 20 : : 1000 : 606, Ans. 
Of a Fluid. 

Rule. — Take a body of known specific gravity, weigh it in and 
out of the fluid ; then, as the weight of the body is to the loss of 
weight, so is the specific gravity of the body to that of the fluid. 

Example. — What is the specific gravity of a fluid in which a piece 
of copper (5. ^.=0000) weighs 70 lbs. in, and 80 lbs. out of if? 
80 : 80—70 : : 9000 : 1125, Ans, 

To find the Quantities of two Ingredients in a Compound, or to 
discover Adulteration in Metals. 

Rule. — Take the differences of each specific gravity of the in- 
gredients and the specific gravity of the compound, then multiply 
the gravity of the one by the difference of the other ; and, as the 
sum of the products is to the respective products, so is the specific 
gravity of the body to the weights of the ingredients. 

Example. — A body compounded of gold {s. ^.=18.888) and silver 
{s. ^.=:10.635) has a specific gravity of 14 : what is the weight of 
each quantity of metal 1 

18.888—14=4.888 X 10.535=51.595 silver, 
14.— 10.535=3.465X18.888=65.447 gold, 

65.447-f 51.495 : 65.447 : : 14 : 7.835 gold, > . 

65.447+51.495 : 51.495 : : 14 : 6.165 silver, ] 

Proof of Spirittums Liquors. 

A cubic inch of proof spirits weighs 234 grains ; then, if an inch 
cube of any heavy body weigh 234 grains less in spirits than air, it 
shows that the spirit in which it was weighed is proof. 

If it lose less of its weight, the spirit is above proof; and if it lose 
more, it is below proof 



The magnitude of a body in cubic feet multiplied by its specific 
gravity, in the following table, gives its weight in avoirdupois 



SPECIFIC GRAVITIES. 



U5 



SOLIDS. 



Divide the Specific Gravity by 16, 

and the quotient is the weight 
of a Cubic Foot in pounds. 



Metals. 

Antimony 

Arsenic , 

Bismuth 

Brass, common , 

Bronze, gun metal 

iJopper, cast 

wire-drawn 

Gold, pure, cast 

hammered 

22 carats fine . 
20 carats fine . 

Iron, cast 

bars 

Lead, cast 

Mercury, 32° 

60O 

Platinum, rolled 

hammered 

Silver, pure, cast 

hammered .. 

Steel, soft 

tempered and hardened 

Tin, Cornish 

Zinc, cast 



Woods (Dry). 

Apple 

Alder 

Ash 

Beech 

Box, Dutch 

French 

Brazilian 

Campeachy 

Cherry 

Cocoa 

Cork 

Cypress 

Ebony, American .... 

Elder ... 

Elm 

Fir, yellow , 

white 

Hacmetac , 

Lignum vitae 

Live Oak , 

Logwood 

Mahogany 

Maple 

Mulberry , 

Oak, English 

heart, 60 years . . . 

Orange 

Pine, yellow 

white 

Poplar 

white 

Pear 

Plum 

Quince • 



SpecifxclY^'^^'^^ 
Graviiy "^ 



6.712 
5.763 
9.823 

7.820 

8.700 

8.788 

8.878 

19.258 

19.361 

17.488 

15.709 

7.20' 

7.788 

11.352 

13.598 

113.580 

22.069 

20.33 

10.474 

10.511 

7.833 

7.818 

7.291 

6.861 



aCu 
bic In. 



Divide the Specific Gravity by 16, 
and the quotient is the weight 
of a Cubic Foot in pounds. 



.793 

.800 

.845 

.852 

.912 

1.328 

1.031 

.913 

.715 

1.040 

.240 

.644 

1.331 

.695 

.671 

.65- 

.569 

.592 

1.333 

1.120 

.913 

1.063 

.750 

.89' 

.932 

1.170 

.705 

.660 

.554 

.383 

.529 

.661 

.785 

.705 

.482 



Lbs. 
.244 

.208 

.355 

.282 

.3 J 5 

.317 

.320 

.697 

.700 

.633 

.563 

.280 

.281 

.410 

.492 

.491 

.798 

.736 

.379 

.381 

.283 

.283 

.263 

.248 



Walnut 

Willow 

Yew, Dutch... 
Spanish - 



.029 

.029 

.031 

.031 

.033 

.048 

.037 

.033 

.026 

.037 

.009 

.023 

.048 

.025 

.024 

.023 

.021 

.021 

.048 

.040 

.033 

.038 

.027 

.032 

.033 

.043 

.025 

.024 

.020 

.014 

.019 

.024 

.029 

.025 

.017 



* Well-seasoned J3m., 1839. 

Ash 

Beech 

Cherry 

Cypress 

Hickory, red 

Mahogany, St. Domingo . 

White Oak, upland 

James River. 

Pine, yellow 

white 

Poplar 

Stones and Earths. 

Alabaster, white 

yellow 

Amber 

Asbestos, starry 

Borax 

Brick 

Chalk 

Charcoal 

triturated 

Clay 

common soil 

Coral, red 

Coal, bituminous 

Newcastle 

Scotch 

Maryland 



Anthracite. 



Diamond 

Earth, loose 

Emery 

Flint, black 

white , 

Gla-ss, flint , 

white 

bottle 

green 

Granite, Scotch 

Susquehanna.. . 

Q-uincy 

Patapsco 

Grindstone — 

Gypsum, opaque 

Hone, white, razor 

Ivory 

Limestone, green 

white 

Lime, quick 

Manganese 

Marble, African 

Egyptian 

Parian 



Specific 
Gravity 



,671 
.585 

.788 



.722 
.624 
.606 
.441 
.838 
.720 
.687 
.759 
.541 
.473 
.587 



2.730 
2.699 
1.078 
3.073 
L714 
1.900 
2.784 
.441 
1.380 
1.930 
1.984 
2.700 
1.270 
1.270 
1.300 
1.355 
1.436 
1.640 
3.521 
1.500 
4.000 
2.582 
2.594 
2.933 
2.892 
2.732 
2.642 
2.625 
2.704 
2.652 
2.640 
2.143 
2.168 
2.876 
1.822 
3.180 
3.156 
.804 
7.000 
2.708 
2.668 
2.838 



Weight 
of a Cu- 
bic In. 

Lbs. 
.024 
.021 
.028 
.029 



.026 
.023 
.022 
.016 
.030 
.026 
.025 
.027 
.020 
.017 
.021 



.099 

.098 

.039 

.111 

.062 

.069 

.100 

.016 

.050 

.070 

.071 

.098 

.046 

.046 

.047 

.049 

.052 

.059 

.127 

.054 

.144 

.094 

.094 

.099 

.098 

.099 

.096 

.095 

.098 

.097 

.096 

.077 

.077 

.104 

.066 

.115 

.114 

.029 

.252 

.098 

.097 

.103 



* Ordnance Manual, 1841. 

N 



146 



SPECIFIC GRAVITIES. 



Table — (Continued) . 



Specific 
Gravity 



Divide the Specific Gravity by 16, 
and the quotient is the weight 
of a Cubic Foot in pounds. 

Stones and Earths. 
Marble, common 

French 

white Italian 

Mica 

Millstone 

Nitre 

Porcelain, China 

Pearl, Oriental 

Phosphorus 

Pumice Stone 

Paving Stone 

Porphyry, red 

Rotten Stone 

Salt, common 

Saltpetre 

Sand 

Shale 

Slate. 

Stone, Bristol 

common 

Sulphur, native 

Tale, black 



Acid, Acetic 

Nitric 

Sulphuric 

Muriatic 

Alcohol, piu-e 

of commerce. 

Ether, sulphuric 

Honey 

Human blood 

Milk 

Oil, Linseed. 



Divide the Specific Gravity by 16, 
and the quolient is the weight 
of a Cubic Foot in pounds. 



G^"''> bicln. 



Miscellaneous. 

Asphaltum 

Beeswax ■ 

Butter 

Camphor 

India rubber 

Fat of Beef 

Hogs.- 

]Mutton 

Gamboge 

Gmipowder, loose - . • • 
shaken.. 

solid 

Gum Arabic 

Indigo 

Lard 

Mastic 

Spermaceti 

Sugar 

Tallow 

Atmospheric air 



Liquids. 



ELASTIC FLUIDS. 



1 cubic foot of atmospheric air 
weighs 527.04 troy grains. 

Its assumed gravity of 1 is the 
unit for elastic fluids. 

Ammoniacal gas 

Azote 

Carbonic acid 

Carburetted hydrogen 

Chlorine • 

Chloro-carbonic 



Specific 
Gravity 



1. 000 
.597 
.976 

1.524 
.555 

2.470 

3.389 



.905 

1.650 

.965 

.942 



Lhs. 
.033 
.058 
.035 
.034 



.988 { .362 

.933; .033 

.923! .033 

.936 1 .034 

.923! .033 

1.222 .044 

.9001 .032 

1.000 i .036 



1.550 
1.800 
1.452 
1.009 

.947 
1.074 

.943 
1.606 

.941 
.0012 



.056 
.065 
.051 
.037 
.0.34 
.038 
.034 
.058 
.034 



Oil, Olive 

Essential, turpentine 

Whale 

Proof Spirit 

Vinegar 

Water, distilled 

sea. 

Dead Sea 

Wine 

Port 

Champagne 



Hydrogen 

Oxygen 

Sulphuretted hydrogen 

*Steam,2120 

Nitrogen 

Vapour of Alcohol 

" of Turpentine spirits • • . 

" of Water 

Smoke of bituminous Coal 

" of Wood 



* Weight of a cubic foot, 25S.3 grains. 



APPLICATION OF THE ABOVE. 

When the Weight of a Body is required. 

Rule.— Find the contents of the body in cubic feet or inches, ana 
multiply it by the factor in the table. 



SPECIFIC GRAVITIES. 



M7 



Example. — What is the weight of a cube of Italian marble, the 
sides being 3 feet 1 

33 X 2708 = 73116 oz. -^16 r= 4569.7 lbs, ^725. 

Or, of a 2 inch sphere of cast iron, 22x.5236x.260 weight of a 
cubic inch =1.089 lbs., Ans. 



Comparative Weight of Timber in a Green and Seasoned Stale. 


Timber 


Weight of a 


Cubic Foot. 




Green. 


Seasoned. 




lbs. OZ. 


lbs. OZ. 


English Oak .... 


71.10 


43. 8 


Cedar 






32. 


28. 4 


Riga Fir . 






48.12 


35. 8 


American Fir 






44.12 


30.11 


Elm . 






66. 8 


37. 5 


Beech 






60. 


53. 6 


Ash . 






58. 3 


50. 



Note. — The average weight of the timber materials in a vessel of war (English) 
is about 50 lbs. to the cubic foot, and for masts and yards about 40 lbs. — Edye's Js". C. 

Given the Diameter of a Balloon to find what Weight it luill raise. 

Rule. — As 1 cubic foot is to the specific difference between at- 
mospheric air, and the gas used to inflate the balloon, so is the ca- 
pacity of the balloon to the weight it will raise. 

Example. — The diameter of a balloon is 26.6 feet, and the gas 
used to inflate it is hydrogen ; what weight will it raise 1 

Sp.gr. of air. Grains. Sp. gr. of hydr. Grains. 

1.000 : 527.04 : : .070 : 36.89 wt. of 1 cubic foot of hydrogen. 
Then 1 : 527.04-^36.89 : : 26.63x.5236 : 4830293. grains, —7000 
(grains in a lb.), =690.04 lbs., Ans. 

Given the Weight to be raised to find the Diameter of a Balloon. 
By inversion of the preceding rule. The weight to be raised is 
690.04 lbs. ; what is the diameter'? 

490.15 ==(527.04 — 36.89) : 1 : : 690.04x7000 : 9854.725 cubic 
feet, 4-. 5236, the cube root of the quotient, is 26.6 feet, Ans. 



148 



STREIS^GTH OF MATERIALS. 



STEENGTH OF MATERIALS, 



COHESION. 



The power of cohesion is that force by which the fibres or parti- 
cles of a body resist separation, and it is therefore proportional ta 
the number of fibres in the body, or to the area of its section. 

Table of the Cohesive Force of Metals, &c. 

Weight or Force necessary to tear asunder 1 Square Inch^ in Avoirdupois 

pounds. 

Metals. 



Copper, cast 

wire 

Gold, cast . 

wire . 

Iron, cast 

wire 
best bar 
medium bar 
inferior bar , 



Gold 5, Copper 1 . 

Brass . 

Copper 10, Tin 1 . 

8, " 1 . 

4, '^ 1 . 



22500 
61200 
20000 
30800 
18000 
50000 
103000 
75000 
60000 
30000 

Compositions. 
50000 
45000 
32000 



Lead, cast . 

milled 
Platinum, wire 
Silver, cast . 
Steel, soft . 

razor . 
Tin, cast block 
Zinc, cast 

sheet . 



36000 
35000 



Silver 5, Copper 1 
" 4, Tin 1 . 
Tin 10, Antimony 1 

" 10, Zinc 1 . 

" 10, Lead 1 . 



Ash . 
Beech . 
Box 

Cedar . 
Chestnut, sweet 
Cj'press 

Deal, Christiana 
Ehu . 
Fir, strongest 
American 
Lance wood 
Lignum vitee 
Locust . 
Mahogany . 



Woods. 
16000 
11500 
20000 
11400 
10500 

6000 
12400 
13400 
12000 

8800 
23000 
11800 
20500 
21000 

Miscellaneous Substances. 



Brick 

Glass plate . . . . 
Hemp fibres glued together 

Ivory 

Marble 



290 

9400 

92000 

16000 

9000 



Mortar, 20 years , 
Slate . 

Stone, fine grain 
Whalebone . 



880 

3320 

53000 

40000 

120 ooa 

150000 

5000 

2600 

16 GOO 



4800!) 
41000 
11000 
12914 
6830 



Mahogany, Spanish 






12000 


Maple . 






10500 


Oak, American white 






11500 


English 






10000 


seasoned 






13600 


Pine, pitch . 






12000 


Norway 






13 COO 


Poplar . 






7000 


Quince 






6000 


Sycamore . 






13000 


Teak, Java . 






1400O 


Walnut 






7800 


Willow 






13000 



52 

12000 

200 

7600 



To find the Strength of Direct Cohesion, 

Rule. — Multiply area of transverse section in inches by the 
weight given in the preceding tables, and the product is the strength 
in lbs. 



STRENGTH OF MATERIALS. 



149 



Example. — In a square bar of ordinary wrought iron, of 2 inches, 
what is the resistance 1 

2X2X60000 r= 240000 lbs., Ans. 

Also, in a rod of cast steel i inch diameter, area of i=.1963x 
120000 = 23556 lbs., Ans. 

The absolute strength of materials, pulled lengthwise, is in propor- 
tion to the squares of their diameters. 



The Lateral or Transverse Strens^th 

Of any beam, or bar of wood, &c., is in proportion to its breadth, 
multiplied by its depth squared, and in like-sided beams as the cube 
of the side of a section. Or, one end being fixed, and the other pro- 
jecting, is inversely as the distance of the weight from the section 
acted upon, and the strain upon any section is directly as the dis- 
tance of the weight from that section. 

The strength of a projecting beam is only one fourth of what it 
would be if supported at both ends, and the weight applied in the 
middle. 

The strength of a projecting beam is only one sixth of one of the 
same length, fixed at both ends, and the weight applied in the mid- 
dle. 

The strength of a beam to support a weight in the centre of it, 
when the ends rest merely upon two supports, compared to one, the 
ends being fixed, is as 2 to 3. 



Tables of the Transverse Strength of Timber. 

AMERICAN. 

One Foot in Lengthy and 1 iTich Square^ Weight suspended from oTie end. 







Breaking 


Greatest 


Weight 


Value for 


Materials. 


weight 


deflexion 


borne with 


general 




in lbs. 


in inches. 


safety. 


use. 


/White Oak 


240 


9. 


196 


72 


?> 


Sweet Chestnut 


170 


1.8 


115 


35 


i 


Yellow Pine 


150 


1.7 


100 


30 


White Pine 


135 


1.4 


95 


32 




Ash 


175 


2.4 


105 


25 


V Hickory .... 


270 


8. 


200 


75 



One Foot in Lengthy and 2 Inches Square. 



White Pine 



1087 



1.5 



800 



32 



Cylinder. On^ Foot in Length. 



White Pine, 2 inches diameter 
White Pine, 1 inch diameter . 



Breaking 
weight 
in lbs. 



610 
75 



N2 



Weight 

borne with 

safety. 



460 
56 



Value for 
general 



20 



150 



STRE^'GTH OF MATERIALS. 



Tables of the Transverse Strength of Cast and Wrought Iron. 

AMERICA ]N\ 

Weight susvcnded from one end,. 

Cylinder. One Foot in Length, and 3 Incfies Dimneter. 



Average of 18 ezperimeats with Gun Metal. 


Breaking 
weight 
in lbs. 


Weight 

borne with 

safety. 


Value for 

general 
use. 


^Cast Iron, cold blast .... 


12000 


10000 


350 



SauARE Bar. One Foot in Length by 2 Inches. 



Gun Metal. 


Breaking 
weight 
in lbs. 


Weight 

borne with 

safety. 


Value for 
general 


Cast Iron, cold blast .... 


5781 


5000 


500 



The values above given are for iron of more than ordinary 
strength; if an inferior article is to be used, a corresponding de- 
duction should be made. 



SauARE Bar. 07ie Foot in Length by 1 Inch. 



Wrought Iron. 


Weight 
borne with 
perfect safety 


Deflexion 
from a hori- 
zontal plane 
without rup- 
ture. 


Weight that 

gave a per- 

manent 

bend. 


Deflexion 
in inches 
with last 
weight. 


Value for 

general 

use. 


t Wrought Iron . . ! 1520 


53<^ 


600 


1 


300 



MISCELLANEOUS. 

Cast /row.— Square bar, side 2 inches, length 12 inches, supported 

at both ends, broke with 22728 lbs. applied in the 

middle. 

Cylinder 3 inches diameter, length 8i inches, broke 

with 17110 lbs. applied at one end. 

White Pme.— Cylinder | ins. diameter, length' 12 inches, broke with 

68 lbs. applied at the end. 
Yellow Fine.—\ inch square, and 15 inches in length, broke with 

125 lbs. applied at the end. 
Hickory and White Oak.— I inch square, and 12 inches in length, re- 
quired 82 lbs. to deflect them i an inch, 
the weight suspended from the end. 
The above and preceding experiments were made by the author in De- 
cember, 1840. 

* From the West Point Foundry Association at Cold Spring, Putnam county, N. Y. 
Specific gravity, 7210. 

t From the Ulster Iron Company, Saugerties, N. Y A fine specimen of ma- 
chinery iron. ^, J 1. I, u 

This specimen hroke with the greatest weight here given, when filed through the 
top to the depth of a i of an inch, and the fracture showed but very httle fibre. 



STRENGTH OF MATERIALS. 151 

Mean Result of several Experiments by English Authors on Cast Iron. 

Square bar, 1 inch by 32, resting upon two supports, broke with 
840 lbs. suspended in the middle. 

Square bar, of 1 inch, projecting 32 inches from a wall, broke with 
278 lbs, applied ; and one, 2 inches deep by i an inch, required 643 lbs. 
to break it. 

Square bar, 1 inch by 32, the ends fixed in walls, required U70 
lbs. suspended from the middle to break it. 

TO FIND THE TRANSVERSE STRENGTH. 

When a Rectangular Bar or Beam is Fixed at on^ End, and Loaded at 

the other. 

Rule.— Multiply the Value in the preceding table by the breadth, 
and square of the depth, in inches, and divide the product by the 
length in feet ; the quotient is the weight in pounds. 

Note.— When the beam is loaded uniformly throughout its length, the result 
must be doubled. 

Example. — What are the weights a cast and a wrought iron bar, 
projecting 30 inches in length, by 2 inches square, will bear'? 
2X22x500-^2.5 = 1600 lbs., Ans. 
2 X 2- X 300H-2.5 = 960 lbs., Ans. 

Or, if the Dimensions of a Beam be required, to support a Given Weight 
at its End. 

^^^^ _WeighO<_len^^ p^^^^^^ of breadth, and square of the 
value in table 
depth. 

Example.— What is the depth of a wrought iron beam, 2 inches 
square, necessary to support 960 lbs. suspended at 30 inches from 
the fixed end '? 

960 X2^ ^ 8, and 8-^2 == 4, and ^4 = 2, Ans. 

When the Bar or Beam is Fixed at both Ends, and Loaded in the 
Middle. 

Rule.— Multiply the Value in the preceding table by six times the 
breadth, and the square of the depth, in inches, and divide by the 
length in feet. 

Note.— When the weight is laid uniformly along its length, the result must be 
tripled. 

Example. — What weight will a bar of cast iron, 2 inches square 
and 5 feet in length, support in the middle, when fixed at the ends 1 
500x6x2x22-^5 = 4800 lbs., Ans. 

When the Bar or Beam is Supported at both Ends, and Loaded in the 

Middle. 

Rule.— Multiply the Value in the preceding table by the square of 



152 STRENGTH OF MATERIALS. 

the depth, and four times the breadth, in inches, and divide the 
product by the length in feet. 

Note.— When the weight is laid uniformly along its length, the result must be 
doubled. 

Example.— What are the weights a cast and a wrought iron bar, 
60 inches between the supports, and 2 inches square, will bear 1 

500 X 2^ X 2 X 4-r-5 — 3200 lbs., Ans. 
300x22x2x^-^5 = 1920 lbs., Ans. 

Or, if the Dimensions be required to Support a Given Weight. 

nuLB.-^^'^^^ ^ ^^"^^^rz: product of four times the breadth, 
value in iable 
and square of the depth. 

Example.— What is the side of a square cast iron beam 2 feet in 
length, between supports, that will support 8000 lbs. in the centre 1 

?2£?^.^4^ = 8, and ^8 = 2.828, Ans. 
500 ^ 

When the Breadth or Depth is required. 

Divide the product obtained by the preceding rules by the square 
of the depth, and you have the breadth; or by the breadth, and the 
square root of the quotient is the depth. 

Example.— If 128 is the product, and the depths, 128-7-82=2, 
the breadth ; 

And -^(128-^2) = 8, the depth. 



When the Weight is n^t in the Middle between the Supports 
Distance from nea rest end x weight ^ ^^^^^^^^ ^p^^ ^^pp^^ 
whole length 
farthest from the weight. 

Distance from farthest end X weight ^^^^„„^^ „^^„ o^r^r^^vf 

. — - — = pressure upon support 

whole length 
nearest the weight. 

When a Beam, supported at both Ends, bears two Weights at unequal 
Distances from the Ends. 

Let D = distance of greatest weight from nearest end, 
d = distance of least weight from nearest end, 
W =: greatest weight, w = least weight, 
L — whole length, / = length from least weight to farthest 

end, 
Z' = distance of greatest weight from farthest end. 



STRENGTH OF MATERIALS. 153 

_, DXW IXw 

Then — | — +- ^ — = pressure at vj end ; 

and -r— -\ — = — = pressure at W end. 

In cylindrical beams or bars, the lateral strength is as the cube ol 
the diameter. 

The strength of a hollow cylinder is to that of a solid cylinder, of 
the same length and quantity of matter, as the greater diameter of 
.the former is to the diameter of the latter ; and the strength of hol- 
low cylinders of the same length, weight, and material, is as their 
greatest diameters. 

To find the Diameter of a Solid Cylinder^ Fixed at both Ends, to 
support a Given Weight in the Middle. 

Rule.— Multiply the length between the supports in feet by the 
weight in pounds ; divide by the value, and the cube root of one 
sixth of the quotient is the diameter in inches. 

Example. — What should be the diameter for a cylinder 2 feet in 
length between the supports, to bear 20000 lbs. 1 

3,20000x2-^350 

v^ ~ = 2.67+, Ans. 

o 

To find the Diameter of a Solid Cylinder, to support a Given 
Weight in the Middle, between the Supports. 

Rule.— Multiply the weight in pounds by the length in feet ; di- 
vide by the Value^ and the cube root of J the quotient is the diameter 
in inches. 

Example. — What is the diameter of a cast iron cylinder, 8 inches 

long between the supports, that will support 60000 lbs. suspended 

in the middle 1 

3 60000 X. 66-^350 ^ ^^ ^ 
^ ^=: 3 03, Ans. 



To find the Diameter of a Solid Cylinder when Fixed at one End, 
the Load applied at the other. 

Rule. — Multiply the length of the projection in feet by the weight 
to be supported in pounds ; divide by the given Value, and the cube 
root of the quotient is the diameter. 

Example. — What should be the diameter of a cast iron cylinder 
8 inches long, to support 15000 lbs. 1 

8 inches is .66 feet, y(15000x.66-f-350) = 3+ inches, Ans. 

Example.— What should be the diameter for 270000 lbs., at 12 
inches from the end 1 

-^(270000 X 1—350) =r: 9.17, Ans. 



254!i STRENGTH OF MATERIALS. 

To find the Diameter of a Beam or Solid Cylinder when the Load 
is uniformly distributed over its Length. 

Rule.— Proceed as if the load was suspended at the end or in the 

middle until the quotient is obtained ; then, , ^ u ir .u:^ 

If for a cylinder with one end fixed, the cube root of half this 

quotient is the diameter ; ^r v^^if ,w^ nnn 

If the ends rest upon two supports, the cube root of half this quo- 
tient is the diameter ; ^ n • •, r ^x,- 

And if the ends are fixed, the cube root of one third of this quo- 
tient is the diameter. 

The Constant Divisor of 350 is for iron of great strength ; where 
an inferior article is to be used, it may be decreased to 250. 

Thus, 350 represents a weight of 9450 lbs. upon the end of a cyl- 
inder 3 inches in diameter and 1 foot in length ad ^50 ^nder the 
same circumstances is equal to a weight of 6750 lbs. 

500 represents a weight of 4000 lbs. upon the end of a bar 2 mches 
square and 1 foot in length, and 400 upon the same bar is equal to 
a weight of 3200 lbs. 

The strength of an equilateral triangle, an edge up, compared to 
a square of the same area, is as 45 to 28. 

To ascertain the Relative Value of Materials to resist a Trans- 
verse Strain. 
T et V represent this value in a beam, bar, or cylinder one foot 
in length and one inch square, side or in diameter. 

1 Fixed at one end. Weight suspended from the other. 

2 Fixed at loth ends: Weight suspended from the middle. 

3 Suvvorted at loth ends. Weight suspended from the middle. 

"~ Ud^ 

4. Su^vorted at hath ends. Weight suspended at any other point 

than the middle. 

mnW 

IbdT' 

5. Fixed at loth ends. Weight suspended at any other point than 

the middle. 

_ 2mnW 

^""m'dF'' 

W representing the weight, I the length & the breadth, ^ the depth, 
m the distance from one end, and n the distance from the other. 



STRENGTH OF MATERIALS. 



155 



From which the value of any of the dimensions may be found, by 
the following formulae : 



\hi^ 



= W 



Ybd' 



W 



7 ^w . .m 



In square beams, &c., b and d = \^"v~ 



^^!I. = w 



ebd^Y 
w 



7 ^^ -h 



IW 



In square beams, &c., b and d = \^^' 



Ud^Y 



= W. 



Ud^Y 
W ' 



IW 



U^Y' 






If/. 



In square beams, &c., Z> and iZ = -5/ 



/i^^V 



mn 



:W 



mTzW - m/zW 



bd'Y ' Id-'Y 



=bW 



4V 

TFT" 



r^. 



In square beams, &c., 5 and d = ^-— ^— . 



-= W . 



2mnW 



= 5. V 



In square beams, &c., b and d = ^ 



,2mnYV _ 
2mnW 



3/V 



When the weight is uniformly distributed, the same formulae will 
apply, W representing only half the required or given weight. 



Mean Results of various Experiments hy English Authors. 



WOODS. 



Fixed at one end. 


Length in 
incli3s. 


Breadth in 
inches. 


Depth in 
inches. 


Breaking 
weight in lbs 


Riga Fir (dry) . 
Riga Fir (wet) . 
Yellow Pine (American) . 
White Pine (Canadian) . 


60 
60 
60 
60 


2 
2 
2 
2 


2 

2 
2 

2 


153 

162 
176 
112 



SOLID AND HOLLOW CYLINDERS. 



Supported at each end. 


Length in 
inches. 


Diameter ex- 
ternal in ins 


Diameter inter- 
nal in inches. 


Deflexion in 
inches. 


Break ins: 
weight in lbs. 


Fir 

Ash . 
Ash 


48 
46 
46 


2 
2 
2 


.5 
1. 


2. 
3. 
3.6 


740 
664 
630 



156 



STRENGTH OF MATERIALS. 



Cast Iron of various Figures^ having equal Sectional Areas. 

Description of bar Distance between sup- 1 Breaking weight 
jjescripiion oi oar. p^^.^^ j^ mcbes. | in Jbs. 


Area of 1 square iucb. 

Square . . . . . 
*' through the diagonal 

2 inches deep by i inch 

3 inches deep by h inch 

4 inches deep by \ inch 

Equilateral Triangles. 

Angle up 

Angle down .... 


36 
32 
32 
32 
32 

32 

32 


897 

851 

2185 

3588 

3979 

1437 
840 



Oak, in seasoning, loses at least ^ of its original weight, and this 
process is facilitated by steaming or boiling. 

It loses more by the former process than the latter. 
By steaming, the specific gravity of a piece of 

oak was reduced from .... 1050 to 744 

By boiling, from 1084 to 788 

By exposure to the air, from .... 1080 to 928 



Weight in air of a cubic foot of 
White Pine, before seasoning . 


Butt. 
Ounces. 

658 


Ounces. 

432 


" " when seasoned . 


549 


416 


Pitch Pine, before seasoning . 
" *' when seasoned . 


628 
540 


597 
529 


Spruce Spar, before seasoning . 
" "■ when seasoned . 


587 
541 


580 
554 


Stiffness of Oak to Cast Iron is as . 




1 to 13 


Strength of Oak to Cast Iron is as . 




1 to 4.5 


Mean Specific Gravity of Yellow Pine 
of Pitch Pine . 


558 

777 




DEFLEXION OF RECTANGULAR 


BEAMS. 





1. The deflexions of the same beam, resting on props at each 
end, and loaded in the middle with weights, are as those weights. 

2. The deflexion is inversely as the cube of the depth ; also, the 
depth being the same, the deflexion is inversely as the breadth. 

3. The deflexion is directly as the cube of the length. 

Let I represent the length of a beam, b its breadth, d its depth, 
and W the weight with which it is loaded ; then the deflexion will 

vary as -rrr ; and if the deflexion is represented by e, then, 

When the Beam is Fixed at one End, and Loaded at the other ^ 

-_- = C, a constant quantity. 
bd^e 

3/3 \Y 
When unifoniuy loaded . g, ,3 = C. 



STRENGTH OF MATERIALS. 
Wh£n SuppaHed at both Ends, aiid Loaded in the Middle, 



157 



32 bd' 



-=zC. 



When uniformly loaded 



5 PW _ 



Hence it follows, that, to preserve the same stiffness in beams, 
the depth must be increased in the same proportion as the length, 
the breadth remaining constant. 

The deflexion of different beams arising from their own weight, 
having their several dimensions proportional, will be as the square 
of either of their like linear dimensions. 

Of three equal and similar beams, one inclined upward, one in- 
clined downward at the same angle, and the other horizontal, it has 
been determined that that which had its angle upward was the 
weakest, the one which declined was the strongest, and the one 
horizontal was a mean between the two. 



Barlow furnishes the following as some of the results obtained by him upon the 
deflexion of beams : 

Length. | Depth. | Breadth. r Lbs. i Deflexion in ins. 


Fir . 
Fir . 
Fir . 


6 feet 
3 " 
6 " 


2 inches 
2 " 


H inches 

2 


180 
, 120 

r 180 


1. 

.10 
2. 



WROUGHT IRON. 

Supported at each end. The average of a number of experiments 
gave, for bars 33 inches in length, 1.9 inches broad, and 2 inches 
deep, a deflexion for every half ton of .024 inches. 

CAST IRON. 

Supported at both ends. Bars 33 inches in length, 1.3 inches in 
breadth, and 0.65 inches deep, deflected 0.27 inches with 162 lbs. 
applied. 

Fir battens. Supported at each end, 15 inches in Jength, and 1 inch 
square, broke with a weight of 440 lbs. ; 30 inches in length, and 1 
inch square, broke with 240 lbs. 

Oak battens. Supported at each end, 2 feet long, H inches deep, and 
I inch m breadth, deflected 1.1 ins., and supported 408 lbs. 

Ash battens. Fixed at one end, 2 feet long, 2 inches deep, and 1 inch 
in breadth, deflected 6 inches with a weight of 434 lbs. 

Fir battens. Fixed at one end, same dimensions as last piece, de- 
flected 3.9 ins. with 276 lbs. 

Note 1. — When a weight is uniformly distributed over the length of a beam, the 
deflexion will be three eighths of the deflexion from the same weight applied at the 
extremity. 

2. If the beam be a cylinder, the deflexion is 1.7 times that of a square beam, 
other tilings being equal. 

3. If the load is unifonnly distributed over the length, the deflexion will be five 
eighths of the deflexion from the same load collected in the middle. 

COHESION. 
In page 148, the results given in the table are those of ultimate* 
resistance ; in practice, i of the weight there given will be suflicient 

O 



158 STRENGTH OF THE JOURNALS OF SHAFTS. 



STRENGTH OF THE JOURNALS OF SHAFTS. 

Wlien tJie Weight is in the Middle of the Shaft. 

Apply the rule under the head of Strength of Materials, and the 
result is the diameter of the journals in inches. 

Example. — What should be the diameter of the journals of a shaft 
lOi feet long to support a wheel of 10,000 lbs. in the centre 1 

Ans. 4.21 ins. 

TO RESIST TORSION. 

Water Wheels^ (^c. 

Rule. — Multiply pressure on the crank pin, or at the pitch line 
of the pinion, by the length of the crank or radius of wheel in feet ; 
divide their product by 125, and the cube root of the quotient is the 
diameter of the journal in inches if of wrought iron. If cast iron is 
to be used, add yq. 

Example. — What should be the diameter for the journal of a wa- 
ter-wheel shaft, the pressure on the crank pin being 594,000 lbs., 
and the crank 5 feet in length ] 

^ == 13.5 inches, Ans. 

Example. — The pressure on a crank pin is 123.680 lbs., and the 
length of crank 5 feet. 

.123680x5 

^ r— =17+, Ans. 

12o 

When tivo Shafts are used, as in Steam Vessels with one Engine, 

^ „, /diameter for one shaft ^x3n ,. ^ . . ,^ 
Rule. — ^{ ) r= diameter m mches. 

Example. — The area of the journal of a single shaft is 113 inches ; 
what should be the diameter if two shafts are used \ ' 
Diameter for area of 113 = 12 inches. 

' A^}^12<1= 10.9, Ans. 
4 

The examples above given are instances in successful practice ; 
where the diameter has been less, fracture has almost universally 
taken place, the strain being increased beyond the ordinary limit. 

Results of Experiments on Torsional Strain. 

Square bars, with a Journal 1 inch in diameter and i inch 271 
length. 

Wrought Iron (Ulster Iron Co.), twisted with 326 lbs., and broke 
with 570 lbs. applied at the end of a lever 30 inches in length. 

Wrought Iron (Swede^), same length of lever, twisted with 367 ' 
lbs., and broke with 615 lbs. 



STRENGTH OF THE JOURNALS OF SHAFTS. 159 

Cast Iron (Foundry), journal 1 inch long, same length of lever, 
broke with 436 lbs. 

The diameters for second and third movers are found by multi- 
plying the diameters ascertained by the above rules by .8 and .793 
respectively. 



Grier, in his Mechanics' Calculator, gives the following rule for 
cast iron shafts : 

240 X number horses' power >. 
-number revolutions per minute>' 
For wrought iron, multiply result by .963, for oak by 2.238, and 
for pine by 2.06. 



„,/ 240 X number horses' powder ^ , . . . , 

^i ■ ; — : ^—. ) = diameter m mches. 

^ ^numbfir revo utions ner mmute/ 



160 GUDGEONS AND SHAFTS. 



GUDGEONS AND SHAFTS. 

To find the Dimensions of a Gudgeon, 
0.30^(wl) = d, 

w representing the stress in 100 lbs., I the length in inches, and d the 
diameter in inches. 

If a Cylindrical Shaft has no other lateral stress to sustain than its 
own weight, and is Fixed at one E7id, 

dz=z. 00002U\ 

Let the stress supposed to be in the middle be n times the weight 
of the shaft ; then. 

When supported at both Ends, 

If the weight of the shaft be not taken i nto account, 

d = ^.0mi2 7d\ 
If the weight of the shaft is taken into account, 

d = ^. 00012 {ni.l)l\ 
When a Hollow Shaft is supported at each End, 

, ^,moisld~~ 

d=V 1 +DS '^ representmg the stress mlbs., / the length 

in inches, D the interior diameter, and d the diameter in inches. 

When a Hollow Shaft is Fixed at each End, and Loaded in the 
Middle, 

, 3 :00048w,/ 
«= V — ^ — +D^- 

For hollow Cylindrical Shafts, supported at one End, 
^ — ^.00048 'i/;/+D3. 

If the hollow shaft support the weights at distances m and n from 
each end, and is supported at each end, 

^=^.00048^i^+D3. 

The last four formulas do not take into account the weight of the shaft. 
The above is for Cast Iron. 

For Cyliiidrical Shafts of Cast Iron to resist Torsion, {Buchanan.) 
Let P be the number of horses' power, and II the revolutions of 
the shaft in a minute ; then 

V _j^ -d. 

For Wrought Iron, multiply this result by .963 ; for Oak, by 2.238 ; 
for Pine, by ^2.06. 

If a shaft has to sustain both lateral stress and torsion, then, 
For cast iron, 

,,/240P wP^ 



TEETH OF WHEELS. 



161 



TEETH OF WHEELS. 

To Construct a Tooth. 

Divide the pitch into 10 parts. Let 3.5 of these parts be below 
the pitch line, and 3.0 of them above. 

The thickness should be 4.7 of the pitch 

The length should be 6.5 of the pitch. 

The Diameter of a wheel is measured from the pitch line. 

The wood used for teeth is about i the strength of cast iron, 
therefore they should be twice the depth to be of equal strength. 

To find the Diameter of a Wheel, the Pitch and Number of Teeth 
being given. 

Pitch X number of teeth 

3:[4i6 = '^'^"'^*^''- 

Note. — The pitch, as found by this rule, is the arc of a circle ; the true pitch 
required is a straight line, and must be measured from the centres of two contigu- 
ous teeth. 



To find the Pitch, the Diameter and Number of Teeth being 
sriven. 



Diameter x 3.1416 
number of teeth 



; pitch. 



To find the Radius. 
Pitch X number of teeth 



3.1416 



2 = radius. 



To find the Number of Teeth. 
2 X radius x 3.1416 



pitch 



: number of teeth. 



Dimensions of Wheels in operation. 



Diameter. 


Breadth. 


Pitch. 


Length of 
teeth. 


Thickness of 
teeth. 


Velocity per 
second. 


Pressure. 


Feet. Ins. 


Inches. 


Inches. 


Inches. 


Inches. 


Feet 


Lbs. 


10 


7. 


2.8 


1.625 


1.3 


3. 


11000 


6 


12. 


4.2 


2.25 


1.9 


6.6 


20000 


7 10 


4.5 


1.9 


1.125 


.875 


1.1 


3300 


14 4 


8. 


3. 


1.75 


1.4 


1.87 


9000 



162 VELOCITY OF WHEELS. 

VELOCITY OF WHEELS. 

The relative velocity of wheels is as the number of their teeth. 

To find the Velocity or Number of Turns of the last Wheel to one 
of the first. 

Rule. — Divide the product of the teeth of the wheels that act as 
drivers by the product of the driven, and the quotient is the number. 

Example. — If a wheel of 32 teeth drive a pinion of 10, on the axis 
of which there is one of 30 teeth, acting on a pinion of 8, what is the 
number of turns of the last 1 

32 30 960 

lo^-s-^W^'"'^"" 

To find the Proportion that the Velocities of the Wheels m a train 
should hear to one another. 

Rule. — Subtract the less velocity from the greater, and divide the 
remainder by one less than the number of wheels in the train ; the 
quotient is the number, rising in arithmetical progression from the 
less to the greater velocity. 

Example. — What are the velocities of three wheels to produce 18 
revolutions per minute, the driver making 3 revolutions per minute 1 

18— 3 = 15 _ then 34-7.5 = 10.5, 

3—1 = 2 ' 

and 10.5+7.5 = 18 ; thus, 3, 10.5, and 18 are the velocities of the 
three wheels. 

To find the Number of Teeth required in a Train of Wheels to 
produce a certain Velocity. 

Rule. — As the velocity required is to the number of teeth in the 
driver, so is the velocity of the driver to the number of teeth in the 
driven. 

Example. — If the driver has 90 teeth, makes 2 revolutions, and 
the velocities required are 2, 10, and 18, what are the number of 
teeth in each of the other two ] 

2d w^heel, 10 : 90 : : 2 : 18 teeth. 
3d wheel, 18 : 90 : : 2 : 10 teeth. 



STRENGTH OF WHEELS. 163 



STRENGTH OF WHEELS. 

The strength of the teeth of wheels is directly as their breadth 
and as the square of their thickness, and inversely as their length. 
The stress is as the pressure. 

To find the Thickness of a Tooth, the Strain at the Pitch Line 
being given. 

Rule.— Divide the pressure in pounds at the pitch line by 3000, 
and the square root of the quotient is the thickness of the tooth in 
inches. 

Example.— The pressure is 9000 lbs., what fs the thickness of the 
tooth required 1 

9000 
"^3000 — ^•'^^^ inches, Ans. 

The Breadth should be 2.5 times the pitch. 

Therefore, as the thickness should be 0.47 of the pitch, the pitch 
for the above example will be 3.685 inches, and the breadth 3.685 
X2.5 = 9.2125 inches. 

To find the Horses'' Power of a Tooth, the Dimensions and 
Velocity being given. 

Thickness ^ x3000 — pressure. 
Pressure x velocity in feet per minute 

33000 = ^''^^^^' P^^^^- 

Thickness x 2.1277+ = the pitch. 
Thickness x 1.5384+ = the length. 

To find the Dimensions of the Arms of a Wheel. 

Rule.— Multiply the power at the pitch line by the cube of the 
length of the arms, and divide this product by the product of the 
number of arms and 280 ; the quotient will be the breadth and cube 
of the depth. 

Example.— If the power be 1600, the diameter of the wheel 10 
feet, and the number of arms 6, what will be the dimensions of each 
arml 

1600X10-^23 200000 

6x280 — ^ ^QQQ = 119 ; if the breadth be 5 inches, then 

119 

— = 23.8, and ^ of 23.8 = 2.87, the depth. 

02 



164 GENERAL EXPLANATIONS CONCERNING WHEELS. 

GENERAL EXPLANATIONS CONCERNING WHEELS. 

Pitch Lines. — The touching circumferences of two or more wheels, which act 
upon each other. 

Pitch of a Wheel. — The distance of two contiguous teeth, measured upon their 
pitch line. 

Length of a Tooth. — ^The distance from its base to its extremity. 

Breadth of a. Tooth. — The length of the face of the wheel. 

Spur Wheels.— Wheels that have their teeth perpendicular to their axis. 

Bevel Wheels. — Wheels having their teeth at an angle with their axis. 

Crown Wheels. — Wheels which have their teeth at a right angle with theL 
axis. 

Mitre Wheels. — Wheels having their teeth at an angle of 45° with their axis. 

Spur Gear. — Wheels acting upon each other in the same plane. 

Bevel Gear. — Wheels acting upon each other at an angle. 

When two wheels act up'on one another, the greater is called the spur or driver ^ 
and the lesser the pinion or driven. 

When the teeth of a wheel are made of a different material from the wheel, they 
are called cogs. 



Table of the Strength of Teeth and Arms. 









Teeth. 




With 6 Arms, 


Pressure in lbs. 


Horses' power 

at 3 feet per 

second. 


Pitch in inches. 


Thickness in 
inches. 


Breadth in 
inches. 


Depth for 1 

foot radius in 

inches. 


Breadth of 
rib in inches. 


22 


.25 


.25 


.119 


.75 


0.87 


.25 


85 


.5 


.50 


.238 


1.25 


1.24 


.42 


191 


1. 


.75 


.357 


1.75 


1.67 


.60 


337 


2. 


1. 


.475 


2.50 


1.76 


.80 


520 


3. 


1.25 


.590 


3. 


2. 


1. 


800 


4. 


1.50 


.730 


4. 


2.20 


1.30 


1040 


5. 


1.75 


.835 


4.25 


2.40 


1.40 


1370 


7. 


2. 


.955 


5. 


2.50 


1.70 


1720 


9. 


2.25 


1.070 


5.50 


2.70 


1.80 


2100 


10.5 


2.50 


1.190 


6. 


2.85 


2. 


2560 


13. 


2.75 


1.310 


6.75 


3. 


2.20 


3000 


15. 


3. 


1.430 


7.25 


3.20 


2.40 


3600 


18. 


3.25 


1.550 


8. 


3.30 


2.60 


4150 


21. 


3.50 


1.670 


8.50 


3.40 


2.80 


4800 


24. 


3.75 


1.790 


9.25 


3.50 


2.90 


5700 


27.5 


4. 


1.910 


10.25 


3.60 


3.40 


6300 


31.5 


4.25 


2.025 


10.50 


3.70 


3. .50 


6900 


34.5 


4.50 


2.150 


11. 


3.80 


3.70 


7700 


38.5 


4.75 


2.270 


11.75 


3.90 


3.90 


8500 


42.5 


5. 


2.390 


12.25 


4. 


4. 



Tredgold. 



HORSE POWER — ANIMAL STRENGTH. 165 



HOESE POWER. 

As this is the universal term used to express the capabihty of first 
movers of magnitude, it is very essential that the estimate of this 
power should be uniform ; and as it is customary, in Europe, to es- 
timate the power of a horse equivalent to the raising of 33000 lbs. 
one foot high in a minute^ there can be no objection to such an esti- 
mate here. 

The estimate, then, of a horse's power in the calculations in this 
work, is 33000 pounds avoirdupois, raised through the space of one 
foot in height in one minute, and in this I am supported by the 
practice of a majority of the manufacturers of steam-engines in 
this country. 



ANIMAL STRENGTH. 



MEN. 



The mean effect of the power of a man, unaided by a machine, 
working to the best possible advantage, and at a moderate estima- 
tion, is the raising of 70 lbs. 1 foot high in a second, for 10 hours in 
a day. 

Two men, working at a windlass at right angles to each other, 
can raise 70 lbs. more easily than one man can 30 lbs. 

Mr. Bevan's results with experiments upon human strength are, for a short pe- 
nod, 



With a drawing-knife 



an auger, both hands 

a screw-driver, one hand 

a bench vice, handle 

a chisel, vertical pressure 

a windlass 

pincers, compression 

a hand-plane . 

a hand-saw 

a thumb-vice . 

a brace-bit, revolving 



a force of 100 lbs. 

100 " 

84 " 

72 " 

72 " 

60 " 

60 " 

50 " 

36 " 

45 " 

16 " 



Twisting by the thumb and fingers only, ) u 
and with small screw-drivers . . j ^^ 

By Mr. Field's experiments in 1838, the maximum power of a strong man, exerted 
for 2^ minutes, is = 18000 lbs. raised one foot in a minute. 

A man of ordinary strength exerts a force of 30 lbs. for 10 hours in a day, with a 
velocity of 2| feet in a second, = 4500 lbs. raised one foot in a minute, = i of the 
work of a horse. 

A foot-soldier travels in 1 minute, in common time, 90 steps, = 70 yards. 

in quick time, 110 " =: 86 " 

in double quick-time, 140 *' = 109 " 
He occupies in the ranks, a front of 20 inches, and a depth of 13, without a knap- 
sack ; the interval between the ranks is 13 inches. 
Average weight of men, 150 lbs. each. 
5 men can stand in a space of 1 square yard. 

A man travels, without a load, on level ground, during 8| hours a day, at the rate 
of 3.7 miles an hour, or 314: miles a day. He can carry 111 lbs. 11 miles in a day. 



166 



ANIMAL STRENGTH. 



A porter going short distances, and returning unloaded, carries 135 lbs. 7 miles a 
day. He can carry, in a wheelbarrow, 150 lbs. 10 miles a day. 
The muscles of the human jaw exert a force of 534 lbs. 

HORSES. 

A horse travels 400 yards, at a walk, in 4-^ minutes ; at a trot, in 
2 minutes ; at a gallop, in 1 minute. 

He occupies in the ranks a front of 40 inches, and a Septh of 10 
feet ; in a stall, from 3^ to 4^ feet front ; and at picket, 3 feet by 9. 

Average weight = 1000 lbs. each. 

K horse, carrying a soldier and his equipments (say 225 lbs.), trav- 
els 25 miles in a day (8 hours). 

A draught horse can draw 1600 lbs. 23 miles a day, weight of car- 
riage included. 

The ordinary work of a horse may be stated at 22.500 lbs., raised 
1 foot in a minute, for 8 hours a day. 

In a horse mill, a horse moves at the rate of 3 feet in a second. The diameter 
of the track should not be less than 25 feet. 

A horse power in machinery is estimated at 33.000 lbs., raised 1 foot in a minute ; 
but as a horse can exert that force but 6 hours a day, one machinery horse power 
is equivalent to that of 4.4 horses. 

The expense of conveying goods at 3 miles per hour per horse teams being 1, the 
expense at 4| miles will be 1.33, and so on, the expense being doubled when the 
speed is 5^ miles per hour. 

The strength of a horse is equivalent to that of 5 men. 



Table ^/ the Amount of Labour a Horse of average Strength is capable 
of performing^ at different Velocities^ on Canals^ Railroads^ and Turrh- 
pikes. 

Force of traction estimated at 83.3 lbs. 



Velocity in miles 


Duration of the 
day's work. 


Useful effect for one day in tons, drawn one mile. 


per hour. 


On a Canal. 


On a Railroad. 


On a Turnpike. 


Miles. 


Hours. 


Tons. 


Tons. 


Tons. 


2i 


Hi 


520 


115 


14 


3 


8 


243 


92 


12 


3^ 


H^ 


153 


82 


10 


4 


^\ 


102 


72 


9. 


5 


2A 


52 


57 


7.2 


6 


2 


30 


48 


6. 


7 


1^ 


19 


41 


5.1 


8 


\\ 


12.8 


36 


4.5 


9 


tV 


9.0 


32 


4.0 


10 


i 


6.6 


28.8 


3.6 



The actual labour performed by horses is greater, but they are injured by it. 



HYDROSTATICS. 167 



HYDROSTATICS. 

Hydrostatics treat of the pressure, weight, and equilibrium of 
non-elastic fluids. 

The pressure of a fluid at any depth is as the depth of the fluid. 

The pressure of a fluid upon the bottom of the containing vessel 
is as the base and perpendicular height, whatever may be the figure 
of the containing vessel. 

Fluids press equally in all directions. 

The Centre of Pressure is that point of a surface against which 
any fluid presses, to which, if a force equal to the whole pressure 
were applied, it would keep the surface at rest. 

The centre of pressure of a parallelogram is at | of the line (meas- 
uring downward) that joins the middles of the two horizontal sides. 

In a triangular plane, when the base is uppermost, the centre of 
pressure is at the middle of the hne, raised perpendicularly from the 
vertex ; and when the vertex is uppermost, the centre of pressure 
is at I of a line let fall perpendicularly from the vertex. 

OF PRESSURE. 

The pressure of a fluid on any surface, whether vertical, ohhque, 
or horizontal, is equal to the weight of a column of the fluid, whose 
base is equal to the surface pressed, and height equal to the distance 
of the centre of gravity of the surface pressed, below the surface of 
the fluid. 

To find the Pressure of a Fluid upon the Bottom of the Contain- 
ing Vessel. 

Rule.— Multiply area of base in feet by height of fluid in feet, and 
their sum by the weight of a cubic foot of the fluid. 

Example. — What is the pressure upon a surface 10 feet square, 
the water (fresh) being 20 feet deep '? 

102 X 20 X 62.5 = 125000 lbs., Ans. 

The side of any vessel sustains a pressure equal to the area of the 
side, multiplied by half the depth. 

The pressure upon an inclined, curved, or any surface, is as the area 
of the surface, and the depth of its centre of gravity below the fluid. 

Example. — What is the pressure upon the sloping side of a pond 
100 feet square, the depth of the pond being 8 feet '? 
1002x^X62.5 = 625000 lbs., Ans. 

Or, on a hemisphere just covered with water, and 36 inches in 
diameter, 

3X3.1416X—X—X62.5=: 662.5, Ans. 

M At 



168 HYDROSTATICS. 

The pressure upon a number of surfaces is found by multiplying 
the sum of the surfaces into the depth of their common centre of 
gravity, below the surface of the fluid. 

CONSTRUCTION OF BANKS. 

A bank, constructed of a given quantity of materials, will just resist the pressure 
of tiie water when the square of its thickness at the base is to the square of its 
perpendicular height, as the weight of a given bulk of water is to the weight of the 
same bulk of the material the bank is made of, increased by twice the aforesaid 
weight of the given bulk of water. 

Thus, if the bank is made of a stone 2 times heavier than water, the thickness of 
the base should be to the height, as 3 to 6. 

If the height, compared to the thickness of the base, be as 10 to 7, stability is al- 
ways ensured, whatever the specific gravity of the material may be. 

The bottom of a conical, pyramidal, or cylindrical vessel, or of one the section of 
which is that of an inverted frustrum of a cone or pyramid, sustains a pressure 
equal to the area of the bottom and the depth of the fluid. 



FLOOD GATES. 

To find the Strain which a Fluid will exert to make it turn upon 
its Hinges, or open. 

Rule. — Multiply \ of the square of the height by the square of the 
breadth, and take a bulk of water equal to the product. 

Example. — If the gate is 6 feet square, 

_X62 =324 cubic feet, or 20250 lbs. 



To find the Strain the Water exerts upon its Hinges. 

Rule. — Multiply ^ of the breadth by the cube of the height, and 
take a bulk of water equal to the product. 

Example. — With the same gate, 

^ X 63 := 216 cubic feet, or 13500 lbs. 



PIPES. 

To find the Thickness of a Pipe. 

RuLE.T— Multiply the height of the head of the fluid in feet by the 
diameter of the pipe in inches, and divide their product by the co- 
hesion of one square inch of the material of which the pipe is com- 
posed. 

By experiment it has been found that a cast iron pipe, 15 inches in diameter, and 
% of an inch thick, will support a head of water of 600 feet ; and that one of oak, 
of the same diameter, and 2 inches thick, will support a head of 180 feet. 

The cohesive power of cast iron, then, would be 12,000 lbs. ; of oak, 1350 lbs. 

That of lead is 750 lbs. ; and wrought iron boiler plates, riveted together ^ is from 
as to 30,000 lbs. 



HYDROSTATICS. 



169 



In conduit pipes, lying horizontal, and made of lead, their thickness, compared 
to their diameter, should be. 

As 2|, 3, 4, 5, 6, 7, 8 lines, 
To 1, Ih 2, 3, 4^, 6, 7 inches. 

And when made of iron, 

As 1, 2, 3, 4, 5, &c., lines, 
' To 1, 2, 4, 6, 8, &c., inches. 
The tenacity of lead is increased to 3000 by the addition of 1 part of zinc in 8. 



HYDROSTATIC PRESS. 

To find the Thickness of the Metal to resist a Given Pressure. 

Let ;? = pressure per square inch in pounds, r = radius of cylin- 
der, and c = cohesion of the metal per square inch. 

Then -^ = thickness of metal. 

The cohesive force of a square inch of cast iron is frequently estimated 
nt 18000 lbs. 

P 



170 HYDRAULICS AND HYDRODYNAMICS. 



HYDRAULICS AND HYDRODYNAMICS. 

Hydraulics treats of the motion of non-elastic fluids, and Hy- 
drodynamics of the force with which they act. » 

Descending water is actuated by the same laws di^ falling bodies. 
Water will fall through 1 foot in i of a second, 4 feet in i of a 
second, and through 9 feet in | of a second, and so on. 

The velocity of a fluid, spouting through an opening in the side 
of a vessel, reservoir, or bulkhead, is the same that a body would 
acquire by falling through a perpendicular space equal to that he 
tween the top of the water and the middle of the aperture. 
Then, by rule 4 in Gravitation, 

^ height X 64.33 = velocity. 
Example.— What is the velocity of a stream issuing from a head 
of 10 feet 1 • 

V^ 10 X 64.33 = 25.36 feet, ^ . 
Or, V10X8 == 25.30 feet, ) ^'^''' 
If the velocity be 50.72 feet per second, what is the head] 
50.722-^64.33=140 feet, ) 
Or, 50.72 -^8^ = 40.2 feet, S 
This would be true were it not for the effect of friction, which m 
pipes and canals increases as the square of the velocity. 

The mean velocity of a number of experiments gives 5.4 feet for a 
height of one foot. The theoretical velocity is {^/&^) 8. 

OF SLUICES. 

To find the Quantity of Water which will flow out of an Opening. 

Rule.— Multiply the square root of the depth of the water by 5.4; 
the product is the velocity in feet per second. This, multiplied by 
the area of the orifice in feet, will give the number of cubic feet per 
second. 

Example.— If the centre of a sluice is 10 feet below the surface 
of a pond, and its area 2 feet, what quantity of water will run out 
in one second 1 

-^lOx 5.4X2 = 34. 1496 feet, Ans. 

Note.— If the area of the opening is large compared with the head of the water 
take § of this velocity for the actual velocity. 

OF VERTICAL APERTURES OR SLITS. 

The quantity of water that will flow out of one that reaches as 
high as the surface is § of that which would flow out of the same 
aperture if it were horizontal at the depth of the base. 

Q^^ velocity at bottom X depth x 2 ^ ^^^^^^^ ^^ ^^^ ^ ^^^^^^ 

O 

of cubic feet per second. 



A- 






a 


,'-<" 

y^ 


o 




/ 




C i5 



HYDRAULICS AND HYDRODYNAMICS. 171 



OF STREAMS OR JETS. 

To find the Distance a Jet will he projected from a Vessel through 
an opening in the Side. 

Rule. — B C will always be equal 
to twice the square root of A O X 
OB. 

If is 4 times as deep below A, as 
fl is, will discharge twice the quan- 
tity of water that will flow from a in 
the same time, because 2 is the 
square root of A o, and 1 is the 
square root of A a. 

Note.— The water will spout the farthest when o is equidistant from A and B ; 
and if the vessel is raised above a plane, B must be taken upon the plane. 

The quantities of water passing through equal holes in the same time are as the 
square roots of their depths. 

Example.— A vessel 20 feet deep is raised 5 feet above a plane ; 
how far will a jet reach that is 5 feet from the bottom] 
^15x10 X 2 = 24.48 feet, Ans. 

When a prismatic vessel empties itself by a small orifice, in the time 
of emptying itself, twice the quantity would he discharged if it were kept 
full by a new supply. 

To find the Vertical Height of a Stream projected from a Pipe. 

Rule.— Ascertain the velocity of the stream by computing the 
quantity of water running or forced through the opening ; then, by 
rule 5 in Gravitation, page 140, find the required height. 

Example.— If a fire-engine discharges 16.8 cubic feet of water 
through a | inch pipe in one minute, how high will the water be 
projected, the pipe being directed vertically] 

1 6.8 X 1728 -^ area of | -finches in a foot -^ seconds in a minute 
= 91.6, or velocity of stream in feet per second ; then, by rule, page 
140, 91.6-7-8 = 11.45, and 11.452 = 131.10 feet, Ans. 

Note.— This rule gives a theoretical result; the result in practice is somewhat 



VELOCITY OF STREAMS. 

In a stream, the velocity is greatest at the surface and in the 
middle of the current. 

To find the Velocity of a River or Brook. 

Rule. — Take the number of inches that a floating body passes 
over in one second in the middle of the current, and extract its 
square root ; double this root, subtract it from the velocity at top, 
and add 1 ; the result will be the velocity of the stream at the bot- 
tom ; and the mean velocity of the stream is equal the velocity at 
the surface — v^ velocity at the si^rface +.5. 



172 HYDRAULICS AND HYDRODYNAMICS. 

Example. — If the velocity at the surface and in the middle of a 
stream be 36 inches per second, what is the mean velocity 1 
^36x2— 36+1 =25, the velocity at bottom. 
36_^36+.5rrr30.5, Ans. 

To find the Velocity of Water running through Pipes. 

Rule. — Divide height of head in inches by length of pipe in inch- 
es, and the square root of the quotient, multiplied by 23.3, will give 
the velocity in inches at the orifice. 

Example. — What is the velocity when the head is 9 feet, the pipe 
24 inches long and 2^ inches bore 1 

,/ 108+-24 X 23.3 = 49.49 inches per second, Ans. 



Quantities of Water discharged from Orifices of various forms^ 
the Altitude being constant^ at 34.642 Inches. 

Cubic inches 
Nature and dimensions of the tubes and orifices. discharged 

in a minute. 

1. A circular orifice in a thin plate, the diameter being 

1.7 inches 10783 

2. A cylindrical tube 1.7 inches in diameter, and 5.117 

inches long 14261 

3. A short conical adjutage, 1.7 inches in diameter . 10526 

4. The same, with a cylinder 3.41 inches long added to 

it 10409 

5. The same, the length of the cylinder being 13.65 inch- 

es long 9830 

6. The same, the length of the cylinder being 27.30 inch- 

es long 9216 

Results prove that the discharge of water through a straight 
cylindrical pipe of an unlimited length may be increased only by al- 
tering the form of the terminations of the pipe, by making the inner 
end of the pipe of the same form as the veyia contracta, and the ex- 
tremity a truncated cone, having its length about 9 times the diam- 
eter of the cylinder or pipe attached, and the aperture at the outlet 
to the diameter of the cylinder as 18 is to 10. 

By giving this form, the discharge is over what it would be by 
the cylinder alone as 24 is to 10. 



"WAVES. 

The undulations of waves are performed in the same time as the 
oscillations of a pendulum, the length of which is equal to the breadth 
of a wave, or to the distance between two neighbouring cavities or 
eminences. 



HYDRAULICS AND HYDRODYNAMICS. 



173 



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174 HYDRAULICS AND HYDEODYNAIMICS. 



GENERAL RULES. 

Discharge by Horizontal Pipes. 

1. The less the diameter of the pipe, the less is the proportional discharge of the 
fluid. 

2. The greater the length of the discharging pipe, the greater the diminution of 
the discharge. Hence, the discharges made in equal times by pipes of different 
lengths, of the same diameter, and under the same altitude of water, are to one 
another in the inverse ratio of the square roots of their lengths. 

3. The friction of a fluid is proportionally greater in small than in large pipes. 
4. The velocity of water flowing out of an aperture is as the square 

root of the height of the head of the water. 

Theoretically the velocity would be -y/ height X8. In practice it is 
^ height X 5.4 = velocity in feet per second. 

Discharge by Vertical Pipes. 
The discharge of fluids by vertical pipes is augmented, on the principle of the 
gravitation of falling bodies ; consequently, the greater the length of the pipe, the 
greater the discharge of the fluid. 

Discharge by Inclined Pipes. 
A pipe which is inclined will discharge in a given time a greater quantity of 
water than a horizontal pipe of the same dimensions. 

Deductions from various Experiments. 

1. The areas of orifices being equal, that which has the smallest perimeter will 
discharge the most water under equal heads ; hence circular apertures are the 
most advantageous. . 

2. That in consequence of the additional contraction of the fluid vem, as the 
head of the fluid increases the discharge is a little diminished. 

3. That the discharge of a fluid through a cylindrical horizontal tube, the diam- 
eter and length of which are equal to one another, is the same as through a simple 

orifice. -, •,. r- v, - 

4. That the above tube may be increased to four times the diameter of the ori- 
fice with advantage. 

5. The velocity of motion that would result from the direct, unretarded ac- 
tion of the column of a fluid which produces it, being a constant, or .8. 

The velochy through an aperture in a thin plate, with the same pressure, is 5. 
Through a tube from two to three diameters in length, projecting outward, 6.5 

Through a tube of the same length, projecting inward 5.45 

Through a conical tube of the form of the contracted vein . . . . 7.9 
Curvilineal and rectangular pipes discharge less of a fluid than rectilineal pipes. 

Discharge from Reservoirs receiving no Supply of Water. 
For prismatic vessels the general law applies, that twice as much would be dis- 
charged from the same orifice if the vessel were kept full during the time which is 
required for emptying itself. 

Discharges from Compound or Divided Reservoirs. 
The velocity in each may be considered as generated by the difference of the 
heights in the two contiguous reservoirs ; consequently, the square root of the dif- 
ference will represent the velocity, which, if there are several orifices, must be 
inversely as their respective areas. 

Discliarge by Weirs and Rectangular Notches. 
The quantity of water discharged is found by taking § of the velocity due to the 
mean height, using 5.1 for the coefficient of tlie velocity. 

Example.— What quantity of water will flow from a pond, over a weir 102 inch- 
es in length by 12 inches deep ? 

ly/^ foot X 5.1 X 8.5 area of weir = 28.9 cubic feet in one second. 



HYDRAULICS AND HYDRODYNAMICS. 



175 



Table of the Rise of Water in Rivers^ occasioned by the erection of 
PierSj 4*c. 



l-J 




Amount of obstruction compared with area 


of section of the river. 




9 


'^£^ 


1 


3 


3 


4 


5 


6 


7 


8 


TTT 


>S!di 


To 


To 


To 


To 


TTT 


10 


To- 


To- 






Feet. 


Faet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


1 


.0157 


.0377 


.0698 


.1192 


.2012 


.3521 


.6780 


1.609 


6.639 


2 


.0277 


.0665 


.123J 


.2102 


.3548 


.6208 


1.196 


2.838 


11.71 


3 


.0477 


.1144 


.2118 


.3618 


.6107 


1.069 


2.0.58 


4.885 


20.15 


4 


.0760 


.1822 


.3372 


.5759 


.9719 


1.701 


3.276 


7.775 


32.07 


5 


.1165 


.2793 


.5168 


.8782 


1.490 


2.607 


5.020 


11.92 


49.15 


6 


.1558 


.3736 


.6912 


1.181 


1.993 


3.487 


6.715 


15.94 


65.75 


7 


.2078 


.4983 


.9221 


1.575 


2.658 


4.651 


8.958 


21.26 


87.71 


8 


.2678 


.6^123 


1.188 


2.030 


3.426 


5.995 


11.54 


27.40 


113.0 


9 


.3359 


.80.54 


1.490 


2.557 


4.296 


7.517 


14.48 


34.36 


141.7 


10 


.4119 


.9877 


1.827 


3.122 


5.268 


9.219 


17.75 


42.14 


173.8 



Velocity of Water in Pipes or Seioers. 

The time occupied in an equal quantity of water through a pipe or sewer of equal 
lengths, and with equal falls, is proportionally as follows: 

In a right line as 90, in a true curve as 100, and in passing a right angle as 140. 

The resistance that a body sustains in moving through a fluid is in proportion to 
the square of the velocity. 

The resistance that any plane surface encounters in moving through a fluid with 
any velocity is equal to the weight of a column whose height is the space a body 
would have to fall through in free space to acquire that velocity, and whose base 
is the surface of the plane. 

Rx AMPLE. — If a plane, 10 inches square, move through water at the rate of 8 
feet per second, then 82-^-64=1.=: the space a body would require to fall to ac- 
quire a velocity of 8 feet per second ; and as 1 foot= 12 inches, then 10X12 =:; 120 
cubic inches, = the column of water whose height and base are required. 

Cub. Inches. Ounces. 

As 1728 : 120 : : 1000 : 69.4, or 4.3 lbs., which is the amount of resistance met 
^vith by the plane at the above velocity. 

And it is the same, whether the plane moves against the fluid or the fluid against 
the plane. 

The following Table shows the results of experiments with a plane one foot 
square, at an immersion of 3 feet below the surface, and at different velocities 
per second. 



Velocity. 


Resistance. 


Velocity. 


Resistance. 


Velocity. 


Re^tance. 


5 feet 

6 " 

7 " 


29.5 lbs. 
40. - 

54.6 " 


8 feet 

9 " 
10 " 


71.7 lbs. 
90.6 " 
112. " 


11 feet 

12 " 
13i " 


136.3 lbs. 
162.1 " 
213. " 



176 WATER WHEELS. 



WATER WHEELS, 

This subject belongs properly to Hydrodynamics, but a separate 
classification is here deemed preferable. 

Water Wheels are of three kinds, viz., the Overshot^ Undershot^ 
and Breast. 

The Overshot Wheel is the most advantageous, as it gives the 
greatest power with the least quantity of water. The next in or- 
der, m point of efficiency, is the Breast Wheel, w^hich may be con- 
sidered a mean betw^een the overshot and the Undershot. For a 
small supply of water w^ith a high fall, th^ first should be employed ; 
where the quantity of w^ater and height of fall are both moderate, 
the second form should be used. For a large supply of water with 
a low fall, the third form must be resorted to. 

Before proceeding to erect a water wheel, the area of the stream 
and the head that can be used must be measured. 

Find the velocity acquired by the water in falling through that 
height by the rule, viz. : Extract the square root of the height of the 
head of the w^ater (from the surface to the middle of the gate), and 
multiply it by 8. 

Note. — Where the opening is small, and the head of water is great, or proper 
tionally so, use from 5.5 to 8 for the multiplier. 

Example. — The dimensions of a stream are 2 by 80 inches, from 
a head of 2 feet to the upper surface of the stream ; what is the ve- 
locity of the w^ater per minute, and what is its w^eight 1 

2 feet and i of 2 inches = 25 inches r= 2.08 feet, v'2.08x*6.5x 
60 z=i 561.60 feet velocity per minute. 

And 80X2X561.6 feet X12 inches, -^1728= 624 cutjic feet, X 
62^ lbs. = 39000 lbs. of w^ater discharged in one minute. 

To find the Power of an Overshot Wheel. 

Rule. — Multiply the weight of water in lbs. discharged upon the 
wheel in one minute by the height or distance in feet from the 
lower edge of the wheel to the centre of the opening in the gate ; 
divide the product by 50000, and the quotient is the number of 
horses' power. 

Example. — In the preceding example, the weight of the water 
discharged per minute is 39000 lbs. If the height of the fall is 23 
feet, the diameter of the wheel being 22, what is the power of the 
wheel ] 

23 feet — 8 inches clearance below = 22.4 = 22.33. 
39000x22.33-^50000=17.41 horses' power, Ans. 

To find the Power of a Stream. 

Rule. — Multiply the weight of the water in lbs. discharged in one 
minute by the height of the fall in feet ; divide by 33000, and the 
quotient is the answer. 

* Estimate of velocity. 



WATER WHEELS. 177 

Example.— What power is a stream of water equal to of the fol- 
lowing dimensions, viz. : 1 foot deep by 22 inches broad, velocity 
350 feet per mmute, and fall 60 feet ; and what should be the size of 
the wheel applied to it ] 

12;< 22X350X12—1728X621X60 feet -f-33000 = 72.9, Ans. 
Height of fall 60 feet, from which deduct for admission of water, 
and clearance below, 15 inches, which gives 58.9 feet for the diam- 
eter of the wheel. 

Clearance above 3 ) , ^ . , 

below 12 I 1^ inches. 

The power of a stream, applied to an overshot wheel, produces 
effect as 10 to 6.6. 

Then, as 10 .- 6.6 : : 72.9 : 48 horses' power equal that of an over- 
shot wheel of 60 feet applied to this stream. 

When the fall exceeds 10 feet, the overshot wheel should be applied, 
the ff ^^^^^^ ^^^ ^^'^^^^ ^^ ^^ proportion to the whole descent, the greater will be 

to^tiie^^^^^ ^s ^s the quantity of water and its perpendicular height multiplied 

The weight of the arch of loaded buckets in pounds, is found by multiplyinff 4 
of their number, X the number of cubic feet in each, and that product by 40. ® 

To find the Power of an Undershot Wheel ivhen the Stream is 
confined to the Wheel. 

Rule.— Ascertain the weight of the water discharged against the 
floats of the wheel in one minute by the preceding rules, and divide 
it by 100000 ; the quotient is the number of horses' power;; 

NoTE.-The 100000 is obtained thus : The power of a stream, applied to an un- 
dershot wheel, produces effect as 10 to 3.3 ; then 3.3 : 10 : : 33000 • 100000 ^ 
.rrn; T ^^^ Opening is nbove the centre of the floats, multiply the weight of the 
water by the height, as m the rule for an overshot wheel. 

Example.— What is the power of an undershot wheel, applied to 
a stream 2 by 80 inches, from a head of 25 feet ? 

\/25x6. 5x60 — 1950 feet velocity of water per minute, and 
2X80 = 160 mches X 1950 X12-M 728 =2166.6 cubic feet X62.5 = 
*135412 lbs. of water discharged in one minute : then 135412— 
100000 = 1.35 horses' power. "^ 

Note.— The maximum work is always obtained when the velocity of the wheel 
IS half that of the stream. Let V represent velocity of float boards, and v velocity 

of water ; then -^^^- X force of the water, will be the force of the efiective stroke. 

V Till l^^i^^ °^ ^^ undershot wheel to the power expended is, at a medium, one 
half that of an overshot wheel. 

The virtual or effective head being the same, the effect will be very nearly as 
the quantity of water expended. 

When the fall is below 4 feet, an undershot wheel should be applied. 

To find the Power of a Breast Wheel. 

Rule.— Find the effect of an undershot wheel, the head of water 
of which is the difference of level between the surface and where it 
strikes the wheel (breast), and add to it the effect of that of an over- 
s hot wheel, the height of the head of which is equal to the diflfer- 

* Equal 160xi2-i-1728x62.5xi950 = momentum of water and its velocity. 



178 WATER WHEELS. 

ence between where the water strikes the wheel, and the tail w^ater ; 
the sum is the effective power. 

Example. — What would-be the power of a breast w^heel applied 
to a stream 2x80 inches, 14 feet from the surface, the rest of the 
fall being 11 feet? 

^14x6,5x60 — 1458.6 feet velocity of water per minute. 

And 2x80x1458x12-^1728 = 1620 cubic feet X 62.5 =:= 101250 
lbs. of water discharged in one minute. 

Then 101250-MOOOOO = 1.012 horses' power as an undershot. 
v^llX6.5x60 = 1290 feet velocity of water per minute. 

And 2x80x1290x12-^1728 rr: 1433 cubic feet X 62.5 = 89562 
lbs. of water discharged in one minute. 

Xll height of fall -^50000= 19.703 horses, which, added to the 
above, =20.715, Ans. 

When the fall exceeds 10 feet, it may be divided into two, and two breast wheels 
applied to it. 

When the fall is between 4 and 10 feet, a breast wheel should be applied. 

The power of a water wheel ought to be taken off opposite to the point where 
the water is producing its greatest action upon the wheel. 



BARKER S MILL. 

The effect of this mill is considerably greater than that which the 
same quantity of water would produce if applied to an undershot 
wheel, but less than that which it would produce if properly applied 
to an overshot wheel. 

Fo7' a description of it, see Griefs Mechanics^ Calculator, page 234. 

Make each arm of the horizontal tube, from the centre of motion to 
the centre of the aperture of any convenient length, not less than ^ of 
the perpendicular height of the w^ater's surface above these centres. 

Multiply the length of the arm in feet by .61365, and the square 
root of the product will be the proper time for a revolution in sec- 
onds ; then adapt the geering to this velocity. Or, if the time of a 
revolution be given, multiply the square of it by 1.6296 for the pro- 
portional length of the arm in feet. 

Divide the continued product of the breadth, depth, and velocity 
of the stream in feet by 14.27 ; multiply the quotient by the square 
root of the height, and the result is the area of either aperture. 

Multiply the area of either aperture by the height of the head ot 
water, and this product by 56 ; the result is the moving force in lbs. 
at the centre of the apertures. 

Example. — If the fall bQ 18 feet from the head to the centre of the 
apertures, then the arm must not be less than 2 feet (as i of 18 = 
2), v/2x.61365 = 1.107, the time of a revolution in seconds; the 
breadth of the race 17 inches, the depth 9, and the velocity 6 feet 
per second ; what is the moving force 1 

17 inches = 1.41 feet, 9 inches = .75 feet; then 1.41 X. 75x6— 
14.27Xx/18xl8x56 = 1895 1bs., .4/i5. 



WATER WHEELS. 179 

To find the Centre of Gyration of a Water Wheel. 

Rule.— Take the radius of the wheel, the weight of its arms, and 
the weight of its rim, as composed of floats, shrouding, &c. 
Let R represent the weight of rim, 
" r '' the radius of the wheel, 
" A " the weight of arms, 

" W " the weight of the water in action when the buck- 
ets are filled, as in operation. 

Then v/(RXr2 X2+A xr- x2+Wxr2-f.R+ATWx2)rr centre ot 
gyration. 

Example.— In a wheel 20 feet diameter, the weight of the rim is 
3 tons, the weight of the arms 2 tons, and the weight of the water 
I ton ; what is the distance of the centre of gyration from the cen- 
tre of the wheel 1 

R =3 tons X10=X2 = 600 

A =2 " Xl02x2=:400 

W=l " X102 . . =100 

3+2+1= 6 X2=:-Y^i= 91.6, the square root of which is 
S.5, or 9i feet, Ans. 

Notes.— At the mill of Mr. Samuel Newlin, at Fishkill Creek, N. Y., 5 barrels 
of flour can be ground, and 400 bushels of grain elevated 36 feet per hour wilh a 
stream and overshot wheel of the following dimensions, viz. : 

Height of head to centre of opening, 24^ inches ; opening, 1% by 80 inches ; wheel, 
22 feet diameter by 8 feet face ; 52 buckets, each 1 foot in depth. 

The wheel making 3^ revolutions, driving 3 run of 5i feet stones 130 turns in a 
mmute, with all the attendant machinery. 

This is a case of maximum effect, in consequence of the gearing being well set 
up, and kept in good order. 

At the furnace of Mr. Peter Townsend, Monroe Works, N. J., 30 to 34 tons of 
No. 1 Iron are made per week, with the blast from two 5 feet by 5 feet 1 inch 
blowing cylinders. The wheel (overshot) being 24 feet diameter, by 6 feet in 
width, having /O buckets of 14 inches in depth. The stream is % by 51 inches 
liaving a head 6^ feet ; the wheel and cylinders each making 4^ revolutions per 

Rocky Glen Factory, Fishkill, N. Y., containing 6144 self-acting mule spindles, 
160 looms, weaving printing cloths 27 inches wide of No. 33 yarn (33 hanks to a 
pound), and producing 24,000 hanks in a day of 11 hours, is driven by a breast 
wheel and stream of the following dimensions, viz. : 

Stream 18 feet by 3 inches, head 20 feet, height of water upon wheel 16 feet, di- 
ameter of wheel 26 feet 4 inches, face of wheel 20 feet 9 inches, depth of buckets 
15f inches, number of buckets 70, 

Revolutions, 4^ per minute. 



180 



PNEUMATICS. 



PNEUMATICS. 



WEIGHT, ELASTICITY, AND RARITY OF AIR. 

The pressure of the air at the surface of the earth is, at a mean 
rate, equal to the support of 29.5 inches of mercury, or 33.18 feet of 
fresh water. It is usually estimated in round numbers at 30 inches 
of mercury and 34 feet of water, or 15 lbs. pressure upon the square 
inch. . . , ,. 

The Elasticity of air is inversely as the space it occupies, and di- 
rectly as its density. 

When the altitude of the air is taken in arithmetic proportion, its 
Rarity will be in geometric proportion. 

Thus, at 7 miles above the surface of the earth, the air is 4 times 
rarer or lighter than at the earth's surface ; at 14 miles, 16 times ; 
at 21 miles, 64 times, and so on. 

The weight of a cubic foot of air is 527.04 grams, or 1.205 ounces 

avoirdupois. j • -nnA 

At the temperature of 33°, the mean velocity of sound is 1100 
feet per second. It is increased or diminished half a foot for each 
degree of temperature above or below 33°. 

To compute Distances hy Sound. 

Rule.— Multiply the time in seconds by 1100, and the product is 
the distance in feet. 

Example.— After observing a flash of lightning, air at 60°, it was 
5 seconds before I heard the thunder ; what was the distance of the 
cloud 1 



1100+- 



50—33 



X 6-^5280 = 1.049 miles, Ans, 



To compute ivhat Degree of Rarefaction may he effected in a 

Vessel, 

Let the quantity of air in the vessel, tube, and pump be represented by 1, and . 
the proportion of 'the capacity of the pump to the vessel and tube by .33 ; conse- 
quently, it contains ^ of the air in the united apparatus. a s c f\.^ r..i 

Upon the first stroke of the piston this fourth will be expelled, and | of the ori- 
ginal quantity will remain : ^ of this will be expelled upon the second stroke, which . 
is equal to ^V of the original quantity ; and, consequently, there remains in the ap- 
paratus ^ of the original quantity. Calculating in this way, the following table ^ 
is easily made : 
No. of stroke*.. 1 Air expelled at each stroke^ I Air remainiDg in the vessel. 



3_ 

16 • 
9^ 

64 

27 
256 

81 
1024 



_ 3 
"~4X4 
3X3 



'4X4X4 

3X3X3 
'4X4X4X4 
_ 3X3X3X3 
'4X4X4X4X4 



16 

27 
64 

?1 
256 
243 
1024 



_3X3 
"4X4 

3X3X3 
"~ 4X4X4 

3X3X3X3 
""4X4X4X4 

3X3X3X3X3 
'4X4X4X4X4 



PNEUMATICS. 



181 



And so on, continually multiplying the air expelled at the preceding stroke by 3 
and dividing it by 4 ; and the air remaining after each stroke is found by multiDlv- 
ing the air remaining after the preceding stroke by 3, and dividing it by 4, 

Measurement of Heights by Means of the Barometer. 
Jlpproximate Rule. For a mean temperature of 550, 
X = required difference in height in feet, 
h^ = the height of the mercury at the lower station, 
h' = the height of the mercury at the upper station, 



X = 55.000 X 



h-\-k' 



■^^^ ^ko °^ ^^^^ result for each degree which the 



mean temperature of the air at the two stations exceeds 550, and deduct as much 
lor each degree below 5oO. *""vi* 



Velocity and Force of Wind. 



Miles in an 
hour. 



Feet in a 
minute. 



1 • 


88 


2 


176 


3 


264 


4 


352 


5 


440 


6 


528 


8 


704 


10 


880 


15 


1320 


20 


1760 


25 


2200 


30 


2640 


35 


3080 


40 


3520 


45 


3960 


50 


4400 


60 


5280 


80 


7040 


100 


8800 



Pressure on a square 
foot in pounds avoir- 
dupois. 



.005 
.020 ) 
.045 ] 
.080 
.125 J 
.180 C 
.320 S 
.500 ) 
1.125 ] 
2.000 ) 
3.125 S 
4.500 ) 
6.125 S 
8.000 ) 
10.125 ] 
12.500 
18.000 
32.000 
50.000 



Description. 



Barely observable. 
Just perceptible. 
Light breeze. 

Gentle, pleasant wind. 

Brisk blow. 
Very brisk. 
High wind. 

Very high. 

Storm. 

Great storm. 

Hurricane. 

Tornado, tearing up trees, &c. 



To find the Force of Wind acting perpendicularly upon a 
Surface. 

RuLE.--Multiply the surface in feet by the square of the velocity 
in feet, and the product by .002288 ; the result is the force in avoir- 
idupois pounds. 

Q 



182 



STATICS. 



STATICS. 



PRESSTJRE OF EARTH AGAINST WALLS. 
A B 




1j I^ ^ . 

Let AB C D be the vertical section of a wall, behind which is a 
bank of earth, AD/.; let DG be the line of rupture, or natural 
slope which the earth would assume but for the resistance of the 
wall. . - 

In sandy or loose earth, the angle G D H is generally 30« ; in firmer 
earth it is 36°, and in some instances it is 45°. 

The angle formed with the vertical by the earth, AD G that ex- 
erts the greatest horizontal stress against a wall, is half the angle 
which the natural slope makes with the vertical. 

If the upper surface of the earth and the wall which supports it 
are both in one horizontal plane. 

Then the resultant In of the pressure of the bank, behind a verti- 
eal wall, is at a distance D w of J A D. 

In ve^retable earths, the friction is J the pressure ; in sands, ^. 

The tine of rupture A G in a bank of vegetable earth is = .618 of A D. 

When the bank is of sand, it is .677 of A D. 

If of rubble, it is .414 of A D. 

Thichiess of Walls, both Faces Vertical. 
Brick. Weight of a cubic foot, 109 lbs. avoirdupois, bank of vegetable earth be- 

^"^Unhewn stones. 135 lbs. per cubic foot, bank as before, A B = .15 A D. 

Brick. Bank clay, well rammed, A B = .17 A D. a « — i-l A n • 

Hewn freestone. 170 lbs. per cubic foot, bank of vegetable earth, A B = .13 A D , 
if the bank is of clay, A B = .14 AD. 

Bricks. Bank of sand, A B = .33 A D. 

Unhewn stone. Bank of sand, A B = .30 A D. 

Hewn freestone. Bank of sand, A B = .26 A D. 

When the bank is liable to be saturated with water, the thickness of the walll 
must be doubled. 

For farther notes, and for the EquUibrium of Piers, see Gregory's Mathematics^ 
pages 220 to 224. 



DYNAMICS. 183 

DYNAMICS. 

Dynamics is the investigation of body^ force, velocity, space, and 
time. 

Let them be represented by their initial letters hfv s t, gravity by 
g, and momentum or quantity of motion by m; this is the effect pro- 
duced by a body in motion. 

Force is motive, and accelerative or retardative. 

Motive force, or momentum, is the absolute force of a body in 
motion, and is the product of the weight or mass of matter in the 
body, multiplied by its velocity. 

Accelerative or retardative force is that which respects the velo- 
city of the motion only, accelerating or retarding it, and is found by 
the force being divided by the mass or weight of the body. Thus, 
if a body of 4 lbs. be acted upon by a force of 40 lbs., the*^ accelera- 
ting force is 10 lbs. ; but if the same force of 40 act upon another 
body of 8 lbs., the accelerating force then is 5 lbs., only half the 
former, and will produce only half the velocity. 

Uniform Motion, 

The space described by a body moving uniformly is represented by the product 
of the velocity into the time. 

With momenta, m varies as b v. 

Example.— Two bodies, one of 20, the other of 10 lbs., are impelled by the same 
momentum, say 69. They move uniformly, the first for 8 seconds, the second for 
6 ; what are the spaces described by both 1 

* 60 o . 60 ^ 

- = .,or- = 3,and- = 6. 

Then «v = 3X8 = 24 = 5, and 6x6 = 36 = ^. 
Thus the spaces are 24 and 36 respectively- 

Motion Uniformly Accelerated. 

In this motion, the velocity acquired at the end of any time whatever, is equal 
to the product of the accelerating force into the time, and the space described is 
equal to the product of half the accelerating force into the square of the time. 

The spaces described in successive seconds of time are as the odd numbers, 1, 3, 
5, 7, 9, &c. 

Grav-ity is a constant force, and its efiect upon a body falling freely is represented 
by^. 

The following theorems are applicable to all cases of motion uniformly accelera- 
ted by any constant force : 

v = Y =gFt =:y/2gfs. 

ts V s 

~~^ -gF -^IgF' 

F— — — -?1 —J^ 
^ gt ~~g^ "^gs 
When gravity acts alone, as when a body falls in a vertical line, F is omitted 
and we have, 

s = igt^ = — = ^tv, 
v = gt =—=^2gs. 



184* 



DYNAMICS. 






^~" £ t2 —05- 

Note.— g is obviously 32.166 from what has been given in rules for Gravitation, 
cuiid is the force of gravity. 

If, instead of a heavy body falling freely, it be propelled vertically upward or 
downward with a given velocity, v, then 

sz^tv::^hgf'\ 
an expression in which — must be taken when the projection is upward, and -j- 
when it is downward. 

Motion over a Fixed Pulleij. 
Let the two weights which are connected by the cord that goes over the pulley 
be represented by W and w ; then 
W — w 



W+w' 



= F in the formulce where F is used ; so that 
W- 



-hgt''' 



Or, if the resistance of the friction and inertia of the pulley be represented by r, 
then 

V7—W „ 

Example. — If by experiment it is ascertained that two weights of 5 and 3 lbs. 
over a pulley, the heavier weight descended only 50 feet in 4 seconds, what is the 
measure of r ? 

If r is not considered, the heavier weight would fall 64^ feet. 

Then „^^~^ ^gt~ — 50 feet. 



And, as 5+3+ r : 5+3 : 
That is . r : 5+3 : 

Whence 



W+MJ+r 
: 64i : 50 ; 
: 14^ : 50. 

_8X14J 



50 



=: 2.293 lbs., Ans. 



Table of the Effects of a Force of Traction of 100 lbs. at different Velo- 
cities J on Canals ^ Railroads^ and Turnpikes. 



Velocity. 


On a Canal. 


On a Railroad. 


On a Turnpike. 


Miles 


Feet per 


JIass 


Useful 


Mass 


Useful 


Mass 


Useful 


perhr. 


second. 


moved. 


etfect. 


moved. 


effect. 


moved. 


effect. 






lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


2^ 


3.66 


55.500 


39.400 


14.400 


10.800 


1.800 


1.350 


3 


4.40 


38.542 


27.361 


14.400 


10.800 


1.800 


1.350 


8^ 


5.13 


28.316 


20.100 


14.400 


10.800 


1.800 


1.350 


4 


5.86 


21.680 


15.390 


14.400 


10.800 


1.800 


1.350 


5 


7.33 


13.875 


9.850 


14.400 


10.800 


1.800 


1.350 


6 


8.80 


9.635 


6.840 


14.400 


10.800 


1.800 


1.350 


7 


10.26 


7.080 


5.026 


14.400 


10.800 


1.800 


1.350 


R 


11.73 


5.420 


3.848 


14.400 


10.800 


1.800 


1.350 


9 


13.20 


4.282 


3.040 


14.400 


10.800 


1.800 


1.350 


10 


14.66 


3.468 


2.462 


14.400 


10.800 


1.800 


1.350 


13.5 


19.9 


U900 


1.350 


14.400 


10.800 


1.800 


1.350 



The load carried, added to the weight of the vessel or carriage which contains it, 
forms the total mass moved, and the useful effect is the load. 

The force of traction on a canal varies as the square of the velocity ; on a rail- 
road or turnpike the force of traction is constant, but the mechanical power neces- 
sary to move the carriage increases as the velocity. 



PENDULUMS. 185 



PENDULUMS. 

The Vibrations of Pendulums are as the square roots of their 
lengths. The length of one vibrating seconds in New- York at the 
level of the sea is 39.1013 inches. 

To find the Length of a Pendulum for any Given Number of 
Vibrations in a Minute. 

Rule. — As the number of vibrations given is to 60, so is the 
square root of 39.1013 (the length of the pendulum that vibrates 
seconds) to the square root of the length of the pendulum required. 
Example. — What is the length of a pendulum that will make 80 
vibrations in a minute? 

As v'39. 1013x60 = 375, a constant number, 
375 
Then _ — 4.6875, and 4.68752 = 21.97 inches, Ans. 

The lengths of pendulums for less ' or greater times is as the 
square of the times ; thus, for i a second it would be the square of 

on lAlO 

h, or — '-- — = 9.7753 inches, the length of a i second pendulum 
at New- York. 

To find the Number of Vibrations in a Minute, the Length of the 
Pendulum being given. 

Rule. — As the square root of the length of the pendulum is to the 
square root of 39.1013, so is 60 to the number of vibrations required. 
Example. — How many vibrations will a pendulum of 49 inches 
long make in a minute \ 

->/49 : v^39.1013 : : 60 : number of vibrations. 
375 
Or, -— = 53.57 vibrations, Ans. 

To find the Length of a Pendulum, the Vibrations of which will 
be the same Number as the Inches in its Length. 

Rule. — Square the cube root of *375, and the product is the an- 
swer. 

Example.— ^375 = 7.211247, and 7.2112472 r= 52.002, Ans. 

The Length of a Pendulum being given, to find the Space through 
which a Body will fall in the Time that the Pendulum makes one 
Vibration. 

Rule.— Multiply the length of the pendulum by 4.93482528, and it 
will give the answer. 

* 375 is the constant for tlie latitude of New-York ; in any other place, multiply 
the square root of the length of the pendulum at that place by 60. 

Q2 



186 CENTRE OF GYRATION. 

Example. — The length of the pendulum is 39.1013 inches ; what is 
the distance a body will fall in one vibration of it'? 

39.1013x4.9348 = 192-9578 inches, or 16.8298 feet, Ans. 

All vibrations of the same pendulum, whether great or small, are performed very 
nearly in the same time. 

In a Simple Pendulum, which is, as a ball, suspended by a rod or line, supposed 
to be inflexible, and without weight, the length of the pendulum is the distance 
from its centre of gravity to its point of suspension. Otherwise, the length of the 
pendulum is the distance from the point of suspension to the Centre of Oscillation.,* 
which does not coincide with the centre of gravity of the ball or bob. 



CENTRE OF GYRATION. 

The Centre of Gyration is the point in any revolving body, or 
system of bodies, that, if the whole quantity of matter were collect- 
ed in it, the angular velocity w^ould be the same ; that is, the mo- 
mentum of the body or system of bodies is centred at this point. 

If a straight bar, equally thick, was struck at this point, the stroke 
would communicate the same angular velocity to the bar as if the 
whole bar w^as collected at that point. 

To find the Centre of Gyration. 

Rule 1. — Multiply the weight of the several particles by the 
squares of their distances in feet from the centre of motion, and 
divide the sum of the products by the weight of the entire mass ; 
the square root of the quotient will be the distance of the centre of 
gyration from the centre of motion. 

Example. — If two weights of 3 and 4 lbs. respectively be laid 
upon a lever (which is here assumed to be without weight) at the 
respective distances of 1 and 2 feet, what is the distance of the 
centre of gyration from the centre of motion (the fulcrum) 1 
3X1'=:3. 4x22zrri6. 

^t^ =-^ = 2.71, and v^2.71 = 1.64 feet, Ans. 

That is, a single weight of 7 lbs., placed at 1.64 feet from the ful- 
crum, and revolving in the same time, would have the same impetus 
as the two weights in their respective places. 

* See Centre of Oscillation. 



CENTRE OF GYRATION. 187 

Rule 2. — Multiply the distance of the centre of oscillation, from 
the centre or point of suspension, by the distance of the centre of 
gravity from the same point, and the square root of the product will 
be the answer. 

Example. — The centre of oscillation is 9 feet, and that of gravity 
is 4 feet from the centre of the system, or point of suspension ; at 
what distance from this point is the centre of gyration 1 
9x4 = 36, and -/36 — 6 feet, Ans. 

The following are the distances of the centres of gyration from 
the centre of motion in various revolving bodies, as given by Mr. 
Farey : 

In a straight, uniform Rodj revolving about one end ; length of rod X-5773. 

In a circular Plate, revolving on its centre ; the radius of the circle X.VOTl. 

In a circular Plate, revolving about one of its diameters as an axis : the radius 
X.5. 

In a Wheel of uniform thickness, or in a Cylinder revolving about the axis ; the 
radius X.7071. 

In a solid Sphere, revolving about one of its diameters as an axis ; the radius 
X.6325. 

In a thin, hollow Sphere, revolving about one of its diameters as an axis ; the 
radius X. 8164. 

In a Cone, revolving about its axis; the radius of the circular base X.5477. 

In a right-angled Cone, revolving about its vertex ; the height of the cone X.866. 

In a Paraboloid, revolving about its axis ; the radius of the circular base X.5773. 

In a straight Lever, the arms being R and r, the distance of the centre of ervra- 

tion from the centre of motion = y/Trr—^- — . 

3(R— r) 

Note. — The weight of the revolving body, multiplied into the height due to the ve- 
locity with which the centre of gyration moves in its circle, is the energy of the body^ 
or the mechanical power which must be communicated to it to give it that motion. 

Example. — In a solid sphere revolving about its diameter, the diameter being 2 
feet, the distance of the centre of gyration is 12x.632o = 7.59 inches. 



188 CENTRES OF PERCUSSION AND OSCILLATION. 



CENTKES OF PERCUSSION AND OSCILLATION. 

The Centres of Percussion and Oscillation being in the same 
point, their properties are the same, and their point is, that in a 
body revolving around a fixed axis, which, when stopped by any 
force, the whole motion, and tendency to motion, of the revolving 
body is stopped at the same time. 

It is also that point of a revolving body v^hich would strike any 
obstacle with the greatest effect, and from this property it has re- 
ceived the name of percussion. 

As in bodies at rest, the whole weight may be considered as col- 
lected in the centre of gravity ; so in bodies in motion, the whole 
force may be considered as concentrated in the centre of percus- 
sion : therefore, the weight of a bar or rod, multiplied by the dis- 
tance of the centre of gravity from the point of suspension, will be 
equal to the force of the rod, divided by the distance of the centre 
of percussion from the same point. 

Example. — The length of a rod being 20 feet, and the weight of a foot in length 
equal 100 oz., having a ball atmched at the under end weighing 1000 oz., at what 
point of the rod from the point of suspension will be the centre of percussion 1* 

The weight of the rod is 20X100 = 2000 oz., which, multiplied bv half its length, 

2000X10 = 20000, gives the momentum of the rod. The weight of the ball = 1000 

oz., multiplied by the length of rod, = 1000X20, gives the momentum of the ball. 

Now the weight of the rod multiplied by the square of the length, and divided by 

2000 V 202 

3, = — -^— — = 268666, the force of the rod, and the weight of the ball multi- 
plied by the square of the lensth of the rod, 1000x20^ = 400000, is the force of the 

V n *-u r .u . / ■ 266666+400000 ^ , ^^ . ^ 

ball : therefore, the centre of percussion = — - — ^^ ^^^^.^ = 16.66 feet. 

' ^ 20000-1-20000 

Example. — Suppose a rod 12 feet long, and 2 lbs. each foot in length, with 2 balls 
of 3 lbs. each, one fixed 6 feet from the point of suspension, and the other at the 
end of the rod ; what is the distance between the points of suspension and percus- 
sion? 

12X 2X6 = 144, momentum of rod, 
3X 6 =18 " of 1st ball, 

3X12 = 36 " of 2d " 

198 
^^^^ = 1152, force of rod, 

3X 36= 108 " of 1st ball, 
3X144 = _432 " of 2d ball, 
1692 
1692 
therefore the centre of percussion = — — = 8.545 feet from the point of suspension* 

luo 

As the centre of percussion is the same with the centre of cscillation in the non-ap- 
plication to practical purposes, the following is the easiest and simplest mode of 
finding it in any beam, bar, &c. : 

Suspend the body very freely by a fixed point, and make it vibrate in small arcs, 
counting the number of vibrations it makes in any time, as a minute, and let the 
number of vibrations made in a minute be called n ; then shall the distance of the 

centre of oscillation from the point of suspension be = — ^ — inches. For the 

length of the pendulum vibrating seconds, or 60 times in a minute, being 39|^ inch- 



'^ ^adXa-j-a 



: 20 feet long, 

-- 100 oz. weight of a foot in lengtli, \ ^""^" ^"^ _ centre of percussion. 

:1000 " fixed at end, ' i-.w.j... 



CENTRES OF PERCUSSION AND OSCILLATION. 189 

es, and the lengths of the pendulums being reciprocally as the square of the num 
ber of vibrations made in the same time, therefore n^ : 60^ : : 39|^ : — - = 

— being the length of the pendulum which vibrates n times in a minute, or 

the distance of the centre of oscillation below the axis of motion. 

There are many situations in which bodies are placed that prevent the applica- 
tion of the above rule, and for this reason the following data are given, which will 
be found useful when the bodies and the forms here given correspond : 

1. If the body is a heavy, straight line of uniform density, and is suspended by 
one extremity, the distance of its centre of percussion is § of its length. 

2. In a slender rod of a cylindrical or prismatic shape, the breadth of which is 
very small compared with its length, the distance of its centre of percussion is 
nearly § of its length from the axis of suspension. 

If these rods were formed so that all the points of their transverse sections were 
equidistant from the axis of suspension, the distance of the centre of percussion 
would be exactly § of their length. 

3. In an Isosceles triangle, suspended by its apex, and vibrating in a plane per- 
pendicular to itself, the distance of the centre of percussion is J of its altitude. A 
line or rod, whose density varies as the distance from its extremity, or the point of 
suspension ; also Fly-wheels, or wheels in general, have the same relation as the 
isosceles triangle, the centre of percussion being distant from the centre of suspen 
sion I of its length. 

4. In a ver>^ slender cone or pyramid, vibrating about its apex, the distance of its 
centre of percussion is nearly f of its length. 

The distance of either of these centres from the axis of motion is found thus : 
If the Axis of Motion be in the vertex of the figure, and the motion be flatwise ; 
then, in a right line, it is § of its length. 
In an Isosceles Triangle = ^ of its height. 
In a Circle = J of its radius. 
In a Parabola t= f of its height. 

But if the bodies move sidewise, it is. 

In a Circle = 5: of its diameter. 

In a Rectangle, suspended by one angle, = § of the diagonal. 
In a Parabola, suspended by its vertex, = -| axis -f- J parameter ; but if suspend* 
ed by the middle of its base, = A axis -}- ^ parameter. 

^ , ^ ^ ^. , 3 X arc X radius 

In the Sector of a Circle = -— — r — 5 

•^ 4 X chord 

^ A . . radius of base ^ 

In a Cone = 4 axis H -— : . 

5 ' 5X axis 

In a Sphere = — r radius + 1, t representing the length of the thread 

5[t X radius) 
by which it is suspended. 

Example.— What must be the length of a rod without a weight, so that when 
hung by one end it shall vibrate seconds 1 

To vibrate seconds, the centre of oscillation must be 39.1013 inches from that of 
suspension ; and as this must be § of the rod, 

Then 2:3:: 39.1013 : 58.6519, Ans. 
Example.— What is the centre of percussion of a rod 23 inches long ? 
§ of 23=15.3 inches from the point of suspension or motion. 
Example. — In a sphere of 10 inches diameter, the thread by which it is suspend- 
ed being 20 inches, where is the centre of percussion or oscillation 1 

These centres are in the same point only when the body is symmetrical with 
regard to the plane of motion, or when it is a solid of revolution, which is com- 
monly the case. 



190 CENTRAL FORCES. 



CENTRAL FORCES. 

All bodies moving around a centre or fixed point have a tendency 
to fly off in a straight line : this is called the Centrifugal Force ; it 
is opposed to the Centripetal Force, or that power which maintains 
the body in its curvilineal path. 

The centrifugal force of a body, moving with different velocities in 
the same circle, is proportional to the square of the velocity. Thus, 
the centrifugal force of a body making 10 revolutions in a minute is 
four times as great as the centrifugal force of the same body making 
5 revolutions in a minute. 

To find the Centrifugal Force of any Body. 

Rule 1. — Divide the velocity in feet per second by 4.01, also the 
square of the quotient by the diameter of the circle ; the quotient 
is the centrifugal force, assuming the weight of the body as 1. 
Then this, multiplied by the weight of the body, is the centrifugal 
force. 

Example. — What is the centrifugal force of the rim of a fly- 
wheel 10 feet in diameter, running with a velocity of 30 feet in a 
second'? 

30—4.01 X7.48-M0=: 5.59 times the weight of the rim, Ans. 

Note. — When great accaracy is required, find the centre of gyration of the body 
and take twice the distance of it from the axis for the diameter. 

Rule 2. — Multiply the square of the number of revolutions in a 
minute by the diameter of the circle in feet, and divide the product 
by the constant number 5870 ; the quotient is the centrifugal force 
when the weight of the body is 1. Then, as in the previous rule, 
this quotient, multiplied by the weight of the body, is the centrifu- 
gal force. 

Example. — What is the centrifugal force of a grindstone, weigh- 
ing 1200 lbs., 42 inches in diameter, and turning with a velocity of 
400 revolutions in a minute 1 

400 2 v*^ 5 

——^-^X 1200= 114480 lbs., Ans. 

The central forces are as the radii of the circles directly, and the squares of the 
times inversely ; also, the squares of the times are as the cubes of the distances. 
Hence, let v represent velocity of body in feet per second, 

w " weight of body, 

r " radius of circle of revolution, 

c " centrifugal force. 

Then -^ =c, and -^ =r; 



, cX32Xr , yrX32Xc\ 
and = 2C, and Vv ) = r. 



CENTRAL FORCES. 191 

Dr. Brewster has famished the following : 

1 The centrifugal forces of two unequal bodies, having the same velocity, and 
at the same distance from the central body, are to one another as the respective 
quantities of matter in the two bodies. 

2 The centrifugal forces of two equal bodies, which perform their revolutions in 
the same time, but are different distances from their axis, are to one another as 
their respective distances from their axis. 

3. The centrifugal forces of two bodies, which perform their revolutions in the 
same time, and whose quantities of matter are inversely as their distances from the 
centre, are equal to one another. 

4. The centrifugal forces of two equal bodies, moving at equal distances from 
the central body, but with different velocities, are t|pone another as the squares of 
their velocities. 

5. The centrifugal forces of two equal bodies, moving with equal velocities at 
different distances from the centre, are inversely as their distances from the centre. 

6. The centrifugal forces of two unequal bodies, moving with equal velocities at 
different distances from the centre, are to one another as their quantities of matter 
multiplied by their respective distances from the centre. 

7. The centrifugal forces of two unequal bodies, having unequal velocities, and 
at different distances from their axis, are, in the compound ratio of their quantities 
of matter, the squares of their velocities, and their distances from the centre. 

The weight of the rim of a fly-wheel for a 20 horse engine is 6000 lbs., the diam- 
eter 16 feet, and the revolutions 45 ; what is its centrifugal force 1 

33120 lbs., Ans. 

Summary.— Let b represent any particle of a body B, and d its distance from the 
axis of motion, S. 
G, O, R, the centres of Gravity, Oscillation, and Gyration, 

Then^ = SG. 
• s*fB = -- 



192 FLY-WHEELS GOVERNORS. 



FLY-WHEELS. 

To find the Weight of Fly-wheels. 

Rule. — Multiply the horses' power of the engine by 2240, and 
divide the product by the square of the velocity of the circumfer- 
ence of the wheel in feet per second ; the quotient will be the 
weight in 100 lbs. 

Example. — The powef'of an engine is 35 horses, the diameter of 
the wheel 14 feet, and the revolutions 40 ; what should be the 
weight of the wheel \ 

35X2240-^40X14X3.1416^602 :=I|^Xl00=r::9130 lbs. 

858.5 

The weight of the wheel in engines that are subjected to irregu- 
lar motion, as in the cotton-press, rolling-mill, &c., must be greater 
than in others where so sudden a check is not experienced, and 3000 
would be a better multiplier in such cases. 



GOVERNORS.' 



The Governor acts upon the principle of central forces. 

When the balls diverge, the ring or the vertical shaft raises, and 
that in proportion to the increase of the velocity squared ; or, the 
square roots of the distances of the ring from the top, corresponding 
to two velocities, will be as these velocities. 

Example. — If a governor make 6 revolutions in a second w^hen 
the ring is 16 inches from the top, w^hat will be the distance of the 
ring when the speed is increased to 10 revolutions in the same 
timel 

As 10'' : 6' : : ^16 inches : 2.4 inches, which, squared, is 5.76 
inches, the distance of the ring from the top. 

A governor performs in one minute half as many revolutions as 
a pendulum vibrates, the length of which is the perpendicular dis- 
• tance between the plane in which the balls move and the centre of 
suspension. 



GUNNERS. 193 



GUNNERY. 

It has been ascertained by experiment that the velocity of the ball 
projected from a gun varies as the square root of the charge direct- 
ly, and as the square root of the weight of the ball reciprocally. — 
Hut ton. 

The same author furnishes the following practical rules : 

To find the Velocity of any Shot or Shell. 

Rule. — As the square root of the weight of the shot is to the 
square root of the weight of treble the weight of the powder, both 
taken in pounds, so is 1600 to the velocity in feet per second. 

Example. — What is the velocity of a shot of 196 lbs., projected 
with a charge of 9 lbs. of powder 1 

14 : 5.2 : : 1600 : 594, Ans. 

When the Range for one Charge is given, to find the Range for 
another Charge, or the Charge for another Range. 

Rule. — The ranges have the same proportion as the charges ; 
that is, as one range is to its charge, so is any other range to its 
charge, the elevation of the piece being the same in both cases. 

Example.— If, with a charge of 9 lbs. of powder, a shot range 4000 
feet, how far will a charge of 6} lbs. project the same shot at the 
same elevation 1 

9 : 6.75 : : 4000 : 3000, Ans. 

Given the Range for one Elevation, to find the Range at another 
Elevation. 

Rule.— As the sine of double the first elevation is to its range, so 
is the sine of double another elevation to its range. 

Example.— If a shot range 1000 yards when projected at an ele- 
vation of 45°, h'ow far will it range when the elevation is 30^ 16', 
the charge of powder being the same 1 

Sine of 45° X 2 =100000, 

Sine of 30° 16^x2=: 87064. 

Then, as 100000 : 1000 : : 87064 : 870.64, Ans. 

Example. — The range of a shell at 45° elevation being 3750 feet, 

at what elevation must a gun be set for a shell to strike an object 

at the distance of 2810 feet with the same charge of powder? 

As 3750 : 100000 : : 2810 : 74934, the sine for double the eleva- 
tion of 240 16', or of 65° 44', Ans. 

R 



194 FRICTION. 



FRICTION. 



Experiments upon the effect of this branch of mechanical science 
are as yet not of such a nature as to furnish deductions for very 

satisfactory rules. ..-u ^ i u • 

The friction of planed woods and polished metals, without luDri- 

cation, upon one another, is about i of the pressure. 

Friction does not increase with the increase of the rubbing sur- 

The friction of metals is nearly constant ; that of woods seems to 
increase with action. . 

The friction of a cylinder rolling upon a plane is as the pressure, 
and inversely as its diameter. .• ^i 

The friction of wheels is as the diameter of their axes directly, 
and as the diameter of the wheel inversely. 

Friction is at a maximum after a state of rest ; the addition is as 
the fifth root of the time. 

The following are the results of some experiments, -without lubrication, as given 
by Adcock : 

FRICTION AFTEH A STATE OF REST. 

At a maximum, oak on oak, ^ to ^^^ of the weight, according to the magnitude 
of the surface ; for oak on pine, 73 ; for pine on pine, — ? for elm on elm, 2Ts of 
the weight, the fibres moving longitudinally. ^ 

When they cross at right angles, the friction of oak is — ; for iron on oak, — ; 
for iron on iron, ^; for iron on brass, \, the surfaces well polished; but when . 

larger, and not so smooth, — . 

For iron on copper, with tallow, the friction is ^ of the weight ; when olive : 
oil is used, the friction is increased to ^. 

The Friction on a level Railroad of a Locomotive is about \ ; that is, an en^ne 
weighing 10 tons has a tractive power of 2 tons by the friction of the surfaces of its 
♦vheels upon the rails. 

FRICTION OF BODIES IN MOTION, 
Without Lubrication. 

When the surfaces are large, the friction increases with velocity. , ^, . , 
For a pressure of from 100 to 4000 lbs. on a square foot, for oak on oak, the fric- 
tion is about —., besides a resistance of about 1§ lbs. for each square foot, independ- 
ent of the pressure. When the surface is very small, the friction is somewhat i 
diminished. For oak on pine, the friction is ~ ; for pine on pine, g ; for iron or i 
copper on wood, ^, which is much increased by an increase of the velocity ; for : 
iron on iron, g^ ; for iron on copper, -^^ ; after much use, ^ at all velocities. 

Where the unctuous matter is interposed between the surfaces, the hardest were 
found to diminish the friction most where the weight was great. Tallow, applied 
between oak, reduced the friction to ^\ of the pressure. When the surfaces arc 
very small, tallow loses its effect, and" the friction is increased to^^^; the adhesion j 
was about 7 lbs. per square foot. 



FRICTION. 195 

With tallow between iron on oak, the friction is 3*^ ; with brass on oak, JL • 

for iron on iron, the friction is y^^, adhesion 1 lb. for 15 square inches ; on copper, 

■y^, adhesion 1 lb. for 13 square inches ; with soft grease or oil, the friction of iron 

on copper and brass was ^ and ^. 

On the whole, in most machines, I of the pressure is a fair estimate of the fric- 
tion. 

FRICTION ON AXES. 

For axes of iron on copper, -^ where the velocity was small, the friction being 
always a little less than for plane surfaces. An axis of iron, with a pulley of giia 
iacum, gave, with tallow, -^^, 

FRICTION AND RIGIDITY OF CORDAGE. 

Wet ropes, if small, are a little more flexible than dry; if large, a little less flexi- 
ble. Tarred ropes are stifler by about ^, and in cold weather somewhat more. 

FRICTION OF PIVOTS. 

When the angle of the summit of the pivot is about IS^ or 20© the friction for 
garnet is y^V"? ^o -^\-^; agate, -g^^; rock crystal, ^|^; glass, ^^ ; and steel 
(tempered), 3^. At an angle of 45° the friction is much reduced, and the friction 
of agate and steel are then nearly equal. 

Notes. — In general, friction is increased in the ratio of the weight. 

Between woods, the friction is less when the grains cross each other than when 
they are placed in the same direction. 

Friction is greater between surfaces of the same kind than between surfaces of 
different kinds. '' 

The best Lubricators are, and in the following order : Tallow, Soft Soao. Lard, 
Oil, and Black-lead. ^' ' 



196 



HEAT. 



HEAT. 

Heat, in the ordinary application of the word, signifies, or, rather, 
implies the sensation experienced upon touching a body hotter, or 
of a higher temperature than the part or parts which we bring into 
contact with it ; in another sense, it is used to express the cause 
of that sensation. 

To avoid any ambiguity that may arise from the use of the same 
expression, it is usual and proper to employ the word Caloric to sig- 
nify the principle or cause of the sensation of heat. 

Caloric is usually treated of as a material substance, though its 
claims to this distinction are not decided ; the strongest argument 
in favour of this position is that of its power of radiation. On 
touching a hot body, caloric passes from it, and excites the feeling 
of warmth; when we touch a body having a lower temperature than 
our hand, caloric passes from the hand to it, and thus arises the 
sensation of cold. 



COMMUNICATION OF CALORIC. 

Caloric passes through different bodies with different degrees of velocity. This 
has led to the division of bodies into conductors and non-conductors of caloric; the 
former includes such as metals, which allow caloric to pass freely through their 
substance, and the latter comprises those that do not give an easy passage to it, 
such as stones, glass, wood, charcoal, &c. 

Table of the relative Conducting Power of different Bodies. 



Platinum . 
Copper 
Zinc . 
Lead 

Porcelain . 
Fire-clay . 

With Water as the Standard. 



Gold . 


. 1000 


Silver 


. 973 


Iron . 


. 374 


Tin . 


. 304 


Marble . 


24 


Fire-brick . 


11 



Water 
Pine . 
Lime 
Oak . 



10 
39 
39 
33 



Elm . 
Ash . 
Apple 
Ebony 



981 

898 

363 

180 
12.2 
11.4 

32 
31 

28 
22 



Relative Conducting Power of different Substances compared with 

each other. 

Hare's fur . . 1.315 Cotton . . . 1.046 

Eider-down . . 1.305 Lint .... 1.032 

Beaver's fur . . 1.296 Charcoal . . . .937 

Raw silk . . . 1.284 Ashes (wood) . . .927 

Wool . . . 1.118 Sewing-silk . . .917 

Lamp-black . . 1.117 Air . . .' . .576 

Relative Conducting Power of Fluids. 

Mercury . . . 1.000 I Proof Spirit . . .312 

Water . , . .357 | Alcohol (pure) . . .232 



Blackened tin . 


100 


Clean "... 


12 


Scraped "... 


16 


Ice 


85 


Mercury .... 


20 


Polished iron . 


15 


Copper .... 


12 



HEAT. 197 

RADIATION OF CALORIC. 

When heated bodies are exposed to the air, they lose portions of their heat, by 
projection in right lines into space, from all parts of their surface. 

Bodies which radiate heat best absorb it best. 

Radiation is affected by the nature of the surface of the body; thus, black and 
rough surfaces radiate and absorb more heat than light and polished surfaces. 

Table of the Radiating Power of different Bodies. 

Water 100 

Lamp-black .... 100 

Writing paper .... 100 

Glass 90 

India ink 88 

Bright lead .... 19 

Silver 12 

Reflection of Caloric is the reverse of Radiation^ and the one increases as 
the other diminishes. 

SPECIFIC CALORIC. 

Specific Caloric is that which is absorbed by different bodies of equal weights 
or volumes when their temperature is equal, based upon the law, acknowledged as 
universal, that similar quantities of different bodies require unequal quantities of 
caloric at any given temperature. Dr. Black termed this, capacity for caloric; but 
as this term was supposed to be suggested by the idea that the caloric present in 
any substance is contained in its pores, and, consequently, the capacities of bodies 
for caloric would be inversely as their densities; and such not being the case, this 
w^ord is apt to give an incorrect notion, unless it is remembered that it is but an ex- 
pression of fact, and not of cause ; and to avoid error, the word specific was propo- 
sed, and is now very generally adopted. 

It is important to know the relative specific caloric of bodies. The most conve- 
nient method of discovering it is by mixing different substances together at differ- 
ent temperatures, and noting the temperature of the mixture ; and by experiments 
it appears that the same quantity of caloric imparts twice as high a temperature 
to mercury as to an equal quantity of water; thus, when water at 100° and mer- 
cury at 40O are mixed together, the mixture will be at 80°, the 20° lost by the water 
causing a rise of 40° in the mercury ; and when weights are substituted for meas- 
ures, the fact is strikingly illustrated ; for instance, on mixing a pound of mercury 
at 40O with a pound of water at 160O a thermometer placed in it will stand at 
1550. Thus it appears that the same quantity of caloric imparts twice as high a 
temperature to mercur>- as to an equal volume of water, and that the heat which 
gives 50 to w^ater will raise an equal weight of mercury 1150, being the ratio of 1 
to 23. Hence, if equal quantities of caloric be added to equal weights of water and 
mercury, their temperatures will be expressed in relation to each other by the 
numbers 1 and 23 ; or, in order to increase the temperature of equal weights of 
those substances to the same extent, the water will require 23 times as much cal- 
oric as the mercury. 

The rule for Ending by calculation, combined with experiment, 
the relative capacities of different bodies, is this : 

Multiply the weight of each body by the number of degrees lost 
or gained by the mixture, and the capacities of the bodies will be 
inversely as the products. 

Or, if the bodies be mingled in unequal quantities, the capacities 
of the bodies will be reciprocally as the quantities of matter, multi- 
plied into their respective changes of temperature. 

The general facts respecting specific caloric are as follows : 

1. Every substance has a specific heat peculiar to itself, whence a change of 
composition will be attended by a change of capacity for caloric. 

R3 



198 



HEAT. 



^ The specific heat of a body varies with its form. A solid has a less capacity 
for caloric than the same substance when in the state of a liquid ; the specific heat 
of water, for instance, being 9 in the solid state, and 10 m the liquid. ^ 

3 The specific heat of equal weights of the same gas increases as-tne density 
decreases ; the exact rate of increase is not known, but the ratio is less rapid than 
the diminution in density. . 

4 Change of capacity for caloric always occasions a €^hange of temperature. In- 
crease in the former is attended by diminution of the latter, and vice versa. 

Tables of the Specific Heat of various Substances. 



Air 

Hydrogen 
Carbonic acid 
Oxygen 
defiant gas . 



1. Air taken as unity. 

Equal volumes. Equal weights. 

1.000 1.000 

.903 12.340 

1.258 .828 

.976 .884 

1.553 1.576 



The specific heat of the foregoing compared with that of an equal 
quantity of water. 



Water 
Air . 
Oxygen 



. 1.000 Hydrogen. . . 3.293 

. 2.669 Carbonic acid . . .221 

. 2.361 defiant gas . . .420 

2. Water taken as unity. 



. .0288 Tellurium . . .0912 

. .0293 Copper . . . .0949 

. .0298 Nickel . . . .1035 

. .0314 Iron 1100 

■ . .0514 Cobalt . •. . .1498 

. .0557 Sulphur . . . .1880 

. .0927 Mercury . . . .0330 

N.— If 1 lb. of coal will heat 1 lb. of water to 100©, -j^ Q) of a 
lb. will heat 1 lb. of mercury to lOOO. 

The term Capacity for heat means the relative powers of bodies, in receiving and 
retaining heat, in being raised to any given temperature ; while Specific applies to 
the actual quantity of heat so received and retained. 

When a body has its density increased, its capacity for heat is diminished. The 
rapid reduction of air to i of its volume evolves heat suflicient to inflame tmder. 

Table showing the relative Capacity for Heat of various Bodies. 



Bismuth 

Lead 

Gold 

Platinum 

Tin . 

Silver 

Zinc 

Illustration.- 





Equal weights. 


Equal vol. 




Equal weights. 


Equal vol. 


Glass 


.187 


.448 


Silver 


.082 


.833 


Iron . 


.126 


.993 


Tin . 


.060 


— 


Brass 


.116 


.971 


Gold . 


.050 


.966 


Copper 


.114 


1.027 


Lead . 


.043 


.487 


Zinc . 


.102 


— 









Latent Caloric is that which is insensible to the touch, or incapable of being 
detected by the thermometer. The quantity of heat necessary to enable ice to 
assume the fluid state is equal to that which would raise the temperature of the 
same weight of water 140° ; and an equal quantity of heat is set free from water 
when it assumes the solid form. 



Ung 



If 5i lbs. of water, at the temperature of 32°, be placed in a vessel, communica- 
ng with another one (in which water is kept constantly boihng at the tempera- 



HEAT. 199 

ture of 2120), until the former reaches this temperature of the latter quantity, then 
let it be weighed, and it will be found to weigh 6^ lbs., showing that 1 lb. of water 
has been received in the form of steam through the communication, and reconvert- 
ed into water by the lower temperature in the vessel. 

Now this pound of water, received in the form of steaip, had, when in that form, 
a temperature of 2120 It is now converted into the liquid form, and still retains 
the same temperature of 212° ; but it has caused 5^ lbs. of water to rise from the 
temperature of 32° to 212^, and this without losing any temperature of itself. It 
follows, then, that in returning to the liquid state, it has parted with 5| times the 
number of degrees of temperature between 32^ and 212^ which are equal 180O 
and 1800x5^ = 9900. Now this lieat was combined with the steam; but as it is 
not sensible to a thermometer, it is called Latent. 

It is shown, then, that a pound of water, in passing from a liquid at 2120 to 
steam at 2120, receives as much heat as would be sufficient to raise it through 990 
thermometric degrees, if that heat, instead of becoming latent, had been Sensible. 

The sum of the Sensible and Latent heat of Steam is always the same 
at any one temperature; thus, 990o+212° == 1202°. 

If to a pound of newly-fallen snow were added a pound of water 
at 172°, the snow would be melted, and 32^ will be the resulting 
temperature. 

Latent Heat of Steam, and several Vapours. 

Steam . 
Alcohol . 
Ether . 

Sensible Caloric is free and uncombined, passing from one sub- 
stance to another, affecting the senses in its passage, determining 
the height of the thermometer, and giving rise to all the results 
which are attributed to this active principle. 

To reduce the Degrees of a Fahrenheit Thermometer to those of 
Reaumur and the Centigrade. 

FAHRENHEIT TO REAtJMUR. 

Rule. — Multiply the number of degrees above or below the free2> 
ing point by 4, and divide by 9. 

Thus, 212°— 32 = 180x4 = 720-^9=80, Ans. 
—24°— 32= 8X4= 32-4-9=3.5,^715. 

FAHRENHEIT TO CENTIGRADE. 

Rule. — Multiply the number of degrees above or below the freez- 
ing point by 5, and divide by 9. 

Thus, 212°— 32 = 180x5 = 900-^9 = 100, Ans. 

Medium Heat of the globe is placed at 50° ; at the torrid zone, 
75° ; at moderate climates, 50° ; near the polar regions, 36°. 

The extremes oi natural heat are from —70° to 120° ; of artificial 
heat, from —91° to 36000°. 



990° 


Nitric acid 


632° 


442° 


Vinegar . 


875° 


302° 


Lead 


610° 



200 



HEAT. 



EVAPORATION. 
Evaporation produces cold, because heat must be absorbed to form vapour. 

Evaporation proceeds only from the surface of the fluids, and therefore ochet 
things equal must depend lipon the extent of surface exposed. 

When a liquid is covered by a stratum of dry air, evaporation is rapid, even 
when the temperature is low. 

As a large quantity of caloric passes from a sensible to a latent state during the 
formation of vapour, it follows that cold is generated by evaporation. 



CONGELATION AND LIQUEFACTION. 

Freezing water gives out 140^ of heat. Water may be cooled to 20°. All soUds 
absorb heat when becoming fluid. 

The particular quantity of heat which renders a substance fluid is called its cal- 
oric of fluidity, or latent heat. 
The heat absorbed in liquefaction is given out again in freezing. 

Fluids boil in vacuo with 124° less of heat than when under the pressure of the at- 
.ere. On Mont Blanc water boils at 187^. 



DISTILLATION. 

Distillation is the depriving vapour of its latent heat, and, though it may be ef- 
fected in vacuo with verv litlle heat, no advantage in regard to a saving of fuel is 
gained, as the latent heat of vapour is increased in proportion to the diminution of 
sensible heat. 

Table of Effects upon Bodies by Heat. 

Chinese porcelain, softened . 
Cast iron, thoroughly smelted 

" " begins to melt 
Smith's forge, greatest heat 
Stone-ware, bakes 
Welding heat of iron (greatest) 

" (least) 

Plate glass, working heat 
Fine gold, melts . 
Fine silver, melts . 
Copper, melts 
Brass, melts . 
Red heat, visible by day 
Iron, red hot in twilight 
Common fire 

Iron, bright red in the dark 
Zinc, melts . 
Quicksilver, boils 
Linseed oil, boils . 
Lead, melts . 
Bismuth, melts 

Tin, melts 

Tin and bismuth, equal parts, melt . 
Tin 3 parts, bismuth 5, and lead 2, melt 
Alcohol, boils 
Ether, boils . 
Human blood (heat of) 
Strong^ines, freeze . 
Brandy, freezes . 
Mercury, melts 
Wedgewood's zero is 1077° of Fahrenheit, 
130O of Fahrenheit. 







Wedzewood. 


Fahrenheit. 


. 1560 


213570 






150O 


205//^ 






130O 


179770 






125<^ 


173270 






102O 


143370 






950 


134270 






90O 


127770 






570 


84870 






320 


52370 






280 


47170 






270 


45870 






210 


3807O 









10770 









8840 









790O 









7520 









700O 









66OO 









6OOO 






\ * 


5940 









4760 






] 


4420 









2830 









2120 









1740 









980 






* 


980 









200 









70 






* _ 


—390 


and 


each 


of his deg 


rees is equal to 



HEAT. 



201 



MISCELLANEOUS. 



FRIGOEIFIC MIXTURES. 



parts 1 



Nitrate of Ammonia 1 part ) 

Water . . . 1 " i 

Phosphate of Soda 9 parts ^ 

Nitrate of Ammonia 6 

Dilute Nitric Acid 4 

Sulphate of Soda 8 parts ; 

Muriatic Acid . 5 " j 

Snow ... 2 parts > 

Muriate of Lime . 3 " i 

Snow ... 8 parts ) 
Dilute Sulphuric Acid 10 " ) 

Snow ... 3 parts ) 

Potash . . . 4 " i 



Thermometer falls, 
or degrees of cold produced. 

460 
710 

5(P 
530 

220 
830 



Degrees of Fahrenheit. 
From + 50O to -f- 4P 

From +500 to— 210 

From +500 to qo 
From— 150 to— 68O 
From— 680 to— 90O 
From 4- 320 to— 510 





EFFECTS OF HEAT. 


Fahrenheit. 


Wedgewood, 




—900 


— 


Greatest cold ever produced. 


—500 


— 


Natural cold at Hudson's Bay. 





— 


Snow and salt, equal parts. 


+430 


— 


Phosphorus burns. 


COO to 770 


— 


Vinous fermentation. 


780 


— 


Acetous fermentation begins. 


88O 


— 


Acetification ends. 


6380 


— 


Lowest heat of ignition of iron in the dark. 


8OOO 


— 


Charcoal burns. 


8490O 


57 


Working heat of plate glass. 


143370 


102 


Stone ware, fired. 


I68O70 


124 


Greatest heat of plate glass. 


251270 


185 


Greatest heat observed. 



EXPANSION OF SOLIDS.. 
At 2120, the length of the bar at 320 considered as 1.0000000. 



Glass 
Platina . 
Cast Iron 
Steel 

" annealed 
Forged Iron . 
Iron wire 



.0008545 
.0009542 
.0011112 
.0011899 
.0012200 
.0012575 
.0014410 



Gold 0014950 

Copper 0017450 

Brass 0019062 

Silver 0020100 

Tin 0026785 

Lead 0028436 

Zinc 0029420 



To find the expansion in Surface, double the above ; in Volume, triple them. 



Table of the Expansion of Air by Heat. 

By Mr. Dalton. 

Fahrenheit. Fahrenheit. Fahrenheit. 



320 , 
330 . 
340 , 
350 . 



1000 

, 1002 

1004 

1107 



40O , 1021 

450 1032 



500 . 
550 , 
6OO , 
650 . 
70O , 
750 . 



1043 


8OO 


1055 


850 


1066 


900 


1077 


1000 


1089 


2000 


1099 


2120 



1110 
1121 
1132 
1152 
1354 
1376 



202 



HEAT. 



MELTING POINT OF ALLOYS. 



Lead 2 parts, Tin 3 parts, 


Bismuth 5 parts, melts at . . . 
" 5 " melts at . . . 


2120 
2460 


" 1 " melts at . 


2860 


a 2 u 


" 1 " melts at . 


3360 


Lead 2 parts, "3 || 


melts at . 
" 1 " melts at . 


3340 
3920 


« 2 " "1 " 


common solder melts at . . . 


4750 


« 1 " " 2 " 


soft solder melts at . . . 


36(P 



GUNPOWDER. 



203 



GUNPOWDER. 



PROPORTIONS OF INGREDIENTS. 

In the United States. Saltpetre. Charcoal. 

Military service .... J ^g] j4[ 

{ 78*. 12*. 

Sporting i 77. 13. 

In England. 

Military service ..... 75. 15. 

o .• S 78. 14. 

Sporting J 75. 17. 

In France. 

Military service 75. 12.5 

o -*• S 78. 12. 

Sporting J 76. 14. 

Blasting 62. 18. 



Sulphur. 
10. 
10. 
10. 
10. 



10. 

8. 

8. 



12.5 
10. 
10. 
20. 



GRANULATION. 

Diameter of sieve holes for Cannon powder . . .070 to .100 inches 

Musket " . . .050 " .070 " 

" " Rifle " . . .025 " .035 " 



DENSITY OF POWDER. 



Size of Grain. 


Specific Gravity. 


Number of 

grains in 10 troy 

grains. 


Weight of 1 cubic foot 


Cubic 


Loose. 


Shaken. 


lb. loose. 


"Cannon 
Musket 
Rifle . . 
Sporting 


1.630 
1.538 
1.535 

1.800 


350 

700 

16.000 

35.000 


oz. 
922 
900 
860 

885 


oz. 

1.000 
990 
960 

1.035 


30 
31 
32 
31 



To find how much Powder will fill a Shell. 
Multiply the cube of the interior diameter in inches by .01744. 
Example. — How much powder will fill a shell, the internal diam- 
eter being 9 inches '? 

93 x.01744= 12.71 lbs., ^715. 



DIMENSIONS OF POWDER BARRELS. 

Whole length " . . 20.5 inches. 

Length, interior in the clear 18. " 

Interior diameter at the head 14. *' 

" " at the bilge 16. " 

Thickness of staves and heads 
Weight of barrels about 

Proof of Powder.— One oz. with a 24 lb. ball, 
at any one time, must not be less than 250 yards ; but none ranging below 225 
yards is received. 

Powder in magazines that does not range over 180 yards is considered unservice- 
able. 

Good powder averages from 280 to 300 yards ; small grain from 300 to 320 yards. 

The greatest initial velocity is obtained by powder of great specific gravity and 
of very coarse grain, giving 130 grains to 10 grains troy. 



25 lbs. 
The mean range of new, proved 



204) LIGHT TONNAGE. 



LIGHT. 

Light is similar to caloric in many of its qualities, bei^g emitted in the form of 
rays, and subject to the same laws of reflection. 

It is of two kinds, J^atural and Artificial; the one proceeding from the sun and 
stars, the other from heated bodies. 

Solids shine in the dark only when heated from 600^ to 700^ and in daylight 
when the temperature reaches iOOOO. 

Relative intensity of light from the burning of various bodies is, for wax, 101 
parts ; tallow, 100 ; oil in an Argand lamp, 110 ; in a common lamp, 129 ; and an 
ill-snuffed candle, 229. 

By experiments on coal gas, it appears that above 20 cubic feet are required to 
produce light equal in duration and in illuminating powers to a pound of tallow 
candles, six to a pound, set up and burned out one after the other. 

In distilling 56 lbs. coal, the quantity of gas produced in cubic feet when the dis- 
tillation was effected in 3 hours was 41.3, in 7 hours 37.5. in 20 hours 33.5, and in 
25 hours 31.7. 



TONNAGE. 

By a law of Congress, the tonnage of vessels is found as follows : 

FOR A DOUBLE-DECKED. 

Take the length from the fore part of the stem to the after side 
of the sternpost above the upper deck ; the breadth at the broadest 
part above the main wales ; half of this breadth must be taken Is 
the depth of the vessel ; then deduct from the length § of the breadth, 
multiply the remainder by the breadth, and the product by the depth ; 
divide this last product by 95, and the quotient is the tonnage. 

Example. — What is the tonnage of a ship of the line, measuring, 
as above, 210 feet on deck, and 59 feet in breath ^ 
59-1-2 = 29.5, depth. 
210 — fof 69 = 174.6X59X29. 5-^95 = 3198.8 tons. 

FOR A SINGLE-DECKED. 

Take the length and breadth as above directed for a double-deck- 
ed, and deduct from the length § of the breadth ; take the depth 
from the under side of the deck-plank to the ceiling of the hold ; then 
proceed as before. 

Example.— -The length of a vessel is (as above) 223 feet, the 
breadth 39i feet, and the depth of hold 23^ feet ; what is the ton- 
nage '? 

223— f of 39.5 =- 199.3 X 39.5 x23.5-^95 ==1947.3 tons. 

A ton will stow 3^ bales cotton. 

Note.— The burden of similar ships are to each other as the cubes of their like 
dimensions. 



TONNAGE. 205 



CARPENTERS' MEASUREMENT. 

FOR A SINGLE-DECKED. 

Multiply the length of keel, the breadth of beam, and the depth of 
the hold together, and divide by 95. 

FOR A DOUBLE-DECKED. 

Multiply as above, taking half the breadth of Ij^am for the depth 
of the hold, and divide by 95. 



To find the Tonnage of English Vessels. 

Rule. — Divide the length of the upper deck between the afterpart of the stem 
and the forepart of the sternpost into 6 equal parts, and note the foremost, middle, 
and aftermost points of division. Measure the depths at these three points in feet, 
and tenths of a foot, also the depths from the under side of the upper deck to the 
ceiling at the limber strake : or, in case of a break in the upper deck, from a line 
stretched in continuation of the deck. For the breadths, divide each depth into 5 
equal parts, and measure the inside breadths at the following points, viz. : at \ and 
at I from the upper deck of the foremost and aftermost depths, and at | and f from 
the upper deck of the midship depth. Take the length, at half the midship depth, 
from the afterpart of the stem to the forepart of the ^sternpost. 

Then, to twice the midship depth, add the foremost and aftermost depths for the 
5^771 of the depths ; and add together the foremost upper and lower breadths, 3 
times the upper breadth with the lower breadth at tlie midship, and the upper and 
twice the lower breadth at the after division for the sum of the breadths. 

Multiply together the sum of the depths, the sum of the breadths, and the length, 
and divide the product by 3500, which will give the number of tons, or register. 

If the vessel have a poop or half-deck, or a break in the upper deck, measure the 
in^e mean length, breadth, and height of such part thereof as may be included 
within the bulkhead ; multiply these three measurements together, and divide the 
product by 92r4. The quotient will be the number of tons to be added to the result 
as above found. 

For Open 'Vessels. The depths are to be taken from the upper edge of the upper 
strake. ^ 

For Steam Vessels. The tonnage due to the engine-room is deducted from the 
total tonnage calculated by the above rule. 

To determine this, measure the inside length of the engine-room from the fore- 
most to the aftermost bulkhead ; then multiply this length by the midship depth 
of the vessel, and the product by the inside midship breadth at 0.40 of the depth 
from the deck, and divide the final product by 92.4. 

S 



206 PILING OF BALLS AND SHELLS. 



PILING OF BALLS AND SHELLS. 

To find the Number of Balls in a Triangular Pile. 

Rule.— Multiply continually together the number of balls in one 
side of the bottom row, and that number increased by 1 ; also, the 
same number increased by 2 ; \- of the product will be the answer. 

Example.— Wh^t is the number of balls in a pile, each side of the 
base containing 30 balls ] • 

30x31x32-^-6 = 4960, Arts. 

To find the Number of Balls in a Square Pile. 

Rule.— Multiply continually together the number in one side of 
the bottom course, that number increased by 1, and double the same 
number increased by 1 ; ^ of the product will be the answer. 

Example.— How many balls are there in a pile of 30 rows 1 
30 X 31 X 61-^6 = 9455, Arts. 

To find the Number of Balls in an Oblong Pile. 

Rule.— From 3 times the number in the length of the base row 
subtract One less than the breadth of the same ; multiply the re- 
mainder by the same breadth, and the product by one more than the 
same, and divide by 6. 

Example.— Required the number of balls in an oblong pile, the 
numbers in the base r ow b eing 16 an d 7 ] 

16x3— 7^X7x7-t-l~6=:392, Ans. 

To find the Number of Balls in an Incomplete Pile. 
Rule.— From the number in the pile, considered as complete, 
subtract the number conceived to be in the upper pile which is want- 
ing. 



WEIGHT AND DIMENSIONS OF BALLS AND SHELLS. 207 



WEIGHT AND DIMENSIONS OF BALLS AND 
SHELLS. 

The weights of these may be found by the rules in Mensuration ; 
also, in the tables, pages 233, 236, and 255. 

To find the Weight of an Iron Ball from its Diameter. 
An iron ball of 4 inches diameter weighs 8.736 lbs. Therefore, 
^ of the cube of the diameter is the weight, for the weight ol 
spheres is as the cubes of the diameters. 

Example. — What is the weight of a ball 10 inches in diameter 1 
!^ of 102 = 136.5 lbs., ^/25. 

To find the Diameter from the Weight. 

Example. — What is the diameter of an iron ball, its weight being 
99.5 lbs. ] 

v^sli ^ ^^-^ — ^ inches, Ans. 
Or, multiply the cube of the diameter in inches by .1365, and the 
sum is the weight. And divide the weight in pounds by .1365, and 
the cube root of the product is the diameter. 



To find the Weight of a Leaden Ball. 
A leaden ball of 4 inches diameter weighs 13.744 lbs. Therefore, 
^^^ of the diameter is the weight. 

Example. — What is the weight of a leaden ball 10 inches in di- 
ameter 1 

^^ of 103 ^ 214.7 lbs., Ans. 

Inversely, v^ j^ X weight = diameter. 

Or, multiply the cube of the diameter in inches by .2147, and the 
sum is the weight. And divide the weight in pounds by .2147, and. 
the cube root of the product is the diameter. 



To find the Weight of a Cast Iron Shell. 

Multiply the difference of the cubes of the exterior and interior 
diameter in incheS by .1365. 

Example. — What is the weight of a shell having 10 and 8.50 inch- 
es for its diameters "? 

103— 8.53 X. 1365 = 52.6 lbs., ^7w. 



208 WINDING ENGINES. 



WINDING ENGINES. 

In winding engines, for drawing coals, water, &c., out of a pit : 
where it is wanted to give a certain number of revolutions, it is ne- 
cessary to know the diameter of the drum and the thickness of the 
rope. 

Where flat ropes are used, and are wound one part over the other, 
To find the Diameter of the Drum. 

Rule.— Multiply the depth of the pit in inches by the thickness 
of the rope in inches for a dividend. 

Multiply the number of revolutions by 3.1416, and the product by 
the thickness of the rope in inches for a divisor. 

Divide the one by the other, and from the quotient subtract the 
product of the thickness of the rope and the num.ber of revolutions ; 
the remainder is the diameter in inches. 

Example.— If an engine make 20 revolutions, the depth of the pit 
being 600 feet, and the thickness of the rope 1 inch, what is the di- 
ameter of the drumj 

600X12X1-^20X3.1416X1— IX 20 = 94.5 inches, Ans. 

To find the Diameter of the Roll. 

Rule.— To the area of the drum add the area or edge surface of 
the rope, and the diameter of the circle having that area is the di- 
ameter of the roll. 

Example.— What is the diameter of the roll in the preceding ex- 
ample % 

Area of 94.5 = 7013.8+ area of 7200 X 1 = 14213.8, and y/ of this 
sum H-.7854 — 134.5, Ans. 

Or, the radius of the drum is increased the number of the revo- 
lutions, multiplied by the thickness of the rope ; as, -^-f-20xl = 
67.25. 

To find the Number of Revolutions. 

, Rule. — To the area of the drum add the area of the edge surface 
of the rope ; then find by inspection in the table of areas, or by cal- 
culation, if necessary, the diameter that gives the exact area; sub- 
tract the diameter of the drum from this, and divide the remainder 
by twice the thickness of the rope ; the quotient is the number of 
revolutions. 

Example.— The length of a rope is 2600 inches, its thickness 1 
inch, and the diameter of the drum 20 inches. Required the num- 
ber of revolutions. 

Area of 20 + area of rope =314.15+2600 = 2914.15, the diame- 
ter of which is 60.91, and 60.91— 20-MX2 = 20.45 revolutions. 



FRAUDULENT BALANCES. 209 

To find the Place of Meeting of the Ascending and Descending 
Buckets when two or more are used. 

Meetings will always be below half the depth of the pit, and 
To find this Depth, 
Take the circumference of the druni for the length of the first turn ; 
then, to the diameter of the drum add twice the thickness of the 
rope, multiplied by the number of revolutions, less 1, for a diameter, 
and the circumference of this diameter is the length of the last turn ; 
add these two lengths together, multiply their sum by half the num- 
ber of revolutions, and the product is the depth of the pit. 

Example. — The diameter of a drum is 9 feet, the thickness of the 
rope 1 inch, and the revolutions 20 ; what is the depth of the pit, 
and at what distance from the top will buckets meet % 
9x3.1416 =28.27, length of first turn; 

2 V 1 v2n 1 

9+ = 12.166X3.1416 = 38.23, length of last turn ; 

20 
28.27+38.23 X— = 665 feet, or depth of pit. 

2 
At 10 revolutions the buckets will meet. Therefore, add 9 times 
twice the thickness of the rope to the diameter of the drum ; to the 
circumference of this diameter add the length of the first turn, 
multiply their sum by half the number of turns to meetings, and the 
product is the distance from the bottom of the pit at which the 
buckets will meet. 

Q V 1 v2 10 

—iiir — 1.5+9x3.1416+28.27x- =306.25 feet, ^715. 



FRAUDULENT BALANCES. 

In order to detect them, after an equilibrium has been established 
between the weight and the article weighed, transpose them, and 
the weight will preponderate if the article weighed is lighter than 
the weight, and contrariwise. Then, 

To ascertain the True Weighty 

Let the weight which will produce equilibrium after transposition 
be found, and with the former weight be reduced to the same de- 
nomination of weight ; and let the two weights thus expressed be 
multiplied together, and the square root of the product will be the 
true weight. 

Example. — If one weight be 7 lbs., and the other 91, 7x91 = 64, 
and the square root of 64 is 8 ; hence 8 lbs. is the true weight. 

Or, let a = length of longest arm, 1 A = greatest weight, 
h = length of shortest arm, I B = least weight. 

Then Wa = Ab, and W6 = Ba ; multiplying these two equations, 
we have W^aJ =r ABa^ or W^ = AB, and W = ^AB. 

S2 



210 MEASURING OF TIMBER. 



MEASURING OF TIMBER. 

Sawed or hewn timber is measured by the cubic foot. 
The unit of board measure is a superficial foot 1 inch thick. 

To measure Round Timber. 

Multiply the length in inches by the square of \ the mean girth 
in inches, and the product, divided by 1728, will give the contents 
in cubic feet. 

When the length is given in feet, and the girth in inches, divide by 144. 

When all the dimensions are in feet, the product is the content without a division. 

Or, ^^^^ -^144, L the length in feet, and C half the sum of the 
16 
circumferences of the two ends in inches. 

Or, ascertain the contents by the rules in Mensuration of Solids, 
page 82, and multiply by .75734. 

Example.— The girths of a piece of timber are 31.416 and 62.832 
inches, and its length 50 feet ; required its contents. 

31.416+62.832_^^^^^^g ^^^ 11.7812x50-^-144 = 48.1916 cu- 
2 
bic feet, Ans. 

Or, ^^^^^-^^^^144=: 48.1916 cubic feet. 

Or, 103— 203-20— 10X.7854X^= 63.632X.75734 = 48.1916 

o 

cubic feet, Ans. 

To measure Square Timber. 

Multiply the length in inches by the breadth in inches, and the 
product by the depth in feet ; divide by 144, and the quotient is the 
content. 

Note.— When all the dimensions are in feet, omit the divisor of 144. 

BOARD MEASURE. 

Multiply the length by the breadth, and the product is the content. 

Note.— This rule only applies when all the dimensions are in feet. When either 
the length or breadth are given in inches, divide their product by 12; and when 
all the dimensions are in inches, divide it by 144. 

Pine spars, from 10 to 4i inches in diameter inclusive, and spruce 
spars, are to be measured^by taking the diameter, clear of bark, at 
J of their length from the large end. 

Spars are usually purchased by the inch diameter ; all under 4 
inches are considered poles. 

Spruce spars of 7 inches and less should have 5 feet in length for 
every inch diameter. Those above 7 inches should have 4 feet in 
length for every inch diameter. 



STEAM. 



211 



STEAM. 

• Steam, aris/.ng from water at the boiling point, is equal to the 
pressure of the atmosphere, which is in round numbers 15 lbs. on 
the square inch. 

Table of the Expansive Force of Steam, from 212° to 352i°. 

(From experiments of Committee of Franklin Institute.) 
The unit is the atmospheric pressure, 30 inches of mercury. 



Degrees of heat. 


Pressure. 


Decrees of heat. 


Pressure. 


Degrees of heat. 


Pressure. 


212.0 


1. 


298.50 


4.5 


331. o 


7.5 


235.0 


1.5 


304.50 


5. 


336.0 


8. 


250.O 


2. 


310.O 


5.5 


340.50 


8.5 


264.0 


2.5 


315.50 


6. 


345.0 


9. 


275.0 


3. 


321.0 


6.5 


349.0 


9.5 


284.0 


3.5 


326.0 


7. 


352.0 


10. 


291.50 


4. 











Under the pressure of the atmosphere alone, water cannot be heated above the 
boiling point. 

It has already been stated (see Heat) that the sum of sensible and latent heats 
is 1202O, and that 140O of sensible heat becomes latent upon the liquefaction of ice ; 
also, that 1 lb. of water converted into steam at 2120 will heat 5^ lbs. of water at 
320 to 2120, and that the sum is 6^ lbs. of water. 



Table of the Volume of Air and Force of Vapour. 



Temperature. 


Volume of air or 
vapour. 


Force of vapour 
in inches of mer- 
cury. 


Temperature. 


Volume of air or 
vapour. 


Force of vapour 

in inches of mer- 

cury. 


OO 
320 

520 

720 
920 
1120 


1000 
1071 
1123 
1183 
1255 
1354 


.032 
.172 
.401 

.842 
1.629 
2.950 


1320 
1520 
1720 
1920 
2120 


1491 
1689 
1930 

2287 
2672 


5.070 

8.330 

13.170 

20.160 

30. 



To ascertain the Number of Cubic Inches of Water, at any Given 
Temperature, that must be mixed ivith a Cubic Inch of Steam 
to reduce the Mixture to any Required Temperature. 

Rule. — From the required temperature subtract the temperature 
of the water ; then find how often the remainder is contained in the 
given temperature, subtracted from 1202^, and the quotient is the 
answer. 

Example.— The temperature of the condensing water of an engine is 80°, and 
the required temperature lOOO ; what is the proportion of condensing water to that 
evaporated ? 

100— 80-8-1202— 100 = -gQ- = 55.5, Ans, 



212 



STEAM. 



Again, the temperature is 60^, and the required temperature KXP. 

1202— 100-f-(100— 60) = ^ = 27.5, ^ns. 

Or, let w represent temperature of condensing water, t the required teraperaturey 
and h the sum of sensible and latent heats. 
h — t 

Then = water required. 

t — w 

To ascertain the Quantity of Steam required to raise a Given 
Quantity of Water to any Given Temperature. 

Rule. — Multiply the water to be warmed by the difference of temperature be- 
tween the cold water and that to which it is to be raised, for a dividend ; then to 
the temperature of the steam add 990^, and from that sum take the required tem- 
perature of the water for a divisor ; the quotient is the quantity of steam in the 
same terms as the water. 

Example. — What quantity of steam at 212^ will raise 100 cubic feet of water at 

80O to 2120 T: 

100x2120 — 80 

o 9o_j_QQno— o 1 oo ~ •'^^•^ c\i^i\c feet of water formed into steam, occupying (13.3X 

1694) 22586.6 cubic feet of space. 

Table of the Boiling Points corresponding to the Altitudes of the 
Barometer between 26 and 31 Inches. 



Barometer. 


Boiling point. 


Barometer. 


Boiling point. 


Barometer. 


Boiling point. 


26. 
26.5 
27. 
27.5 


204.91O 

205.790 

206.67O 
207.550 


28. 
28.5 
29. 
29.5 


2O8.43O 
209.31O 
210.190 
211.070 


30. 

30.5 

31. 


212.0 

212.880 
213.760 



A cubic inch of water, evaporated under the ordinary atmospheric pressure, is 
converted into 1694 cubic inches of steam, or, in round numbers, 1 cubic foot, and 
gives a mechanical force equal to the raising of 2200 lbs. 1 foot high. 

The Pressure of Steam being given, to find its Temperature. 

Rule. — Multiply the 6th* root of the pressure in inches by 177, and subtract 100 
from the product. 

Example. — If the pressure is 240 inches of mercury, what is the temperature 1 
6/240 = 2.493X177—100=341.61, .dns. 

For sea water, multiply by 177.6 ; when -^^ saturated, by 178.3 ; and when ^^ 
saturated, by 179. 



Table 


of the Density of Steam under 


different Pressures. 


Atmospheres. 




Density. 


Volume. 


Atmospheres. 


Density. 


Volume. 


1 




.00059 


1694 


10 


.00492 


203 


2 




.00110 


909 


12 


.00581 


172 


3 




.00160 


625 


14 


.00670 


149 


4 




.00210 


476 


16 


.00760 


131 


5 




.00258 


387 


18 


.00849 


117 


6 




.00306 


326 


20 


.00937 


106 


8 




.00399 


250 









The volumes are not direct, in consequence of the increase of heat. See obser- 
vations, page 198. 



* See page 118 for rule to find this root. 



STEAM. 



213 



Table of the Expansive Force of Steam in Atmospheres. 



Temperature. 


Pressure in 
atmospheres. 


Temperature. 


Pressure in 
atmospheres. 


Temperature. 


Pressure in 
atmospheres. 


212.0 


1 


331.20 


7 


413.80 


19 


242P 


H 


341. 80 


8 


4I8.50 


20 


250.60 


2 


350.80 


9 


423.0 


21 


264.0 


2i 


359.0 


10 


427.30 


22 


2T7.20 


3 


366.80 


n 


431.40 


23 


285.20 


3^ 


374.0 


12 


435.60 


24 


293.80 


4 


380.60 


13 


438.70 


25 


301. o 


4^ 


387.0 


14 


457.20 


30 


308.O 


5 


392.60 


15 


472.80 


35 


314.40 


5.V 


398.50 


16 


486.60 


40 


320.40 


6 


403.80 


17 


499.10 


45 


326.30 


6i 


409.O 


18 


510.60 


50 



Note. — This table gives results slightly differing from that furnished by the 
Franklin Institute, being about 3.5^ for every 5 atmospheres. 



Table of the Pressure, Specific Gravity, and Weight of a Cubic 
Foot of Steam at different Temperatures. 

Pressure in ins. "Weight of a cub. Spec, gravity, Pressure in ins. Weight of a cub. Spec, gravity, 
of mercury. foot in grains. air being 1. of mercury. foot in grains. air being 1. 



.55 

1. 

2. 

3. 

4. 

7.5 
15. 
22.5 
30. 
35. 
45. 
52.5 
60. 



6.10 
10.70 
20.50 
30. 
39. 
71. 

135. 

196. 

254.70 

292. 

363. 

427. 

483. 



.0115 

.0202 

.0388 

.0568 

.0744 

.134 

.255 

.371 

.484 

.553 

.687 

.810 

.915 



75. 

90. 
105. 
120. 
150. 
180. 
210. 
240. 
270. 
300. 
600. 
900. 
1200. 



593.50 


1.123 


700. 


1.33 


810. 


1.53 


910. 


1.728 


1110. 


2.12 


1317. 


2.5 


1520. 


2.88 


1660. 


3.25 


1910. 


3.61 


2100. 


3.97 


3940. 


7.44 


5670. 


10.75 


7350. 


13.88 



A pressure of 1, 5, 10, 20, 40, and 50 lbs. on a square inch, will raise a mercurial 
gauge respectively 1.01, 5.08, 10.16, 20.32, 40.65, and 50.80 inches. 

The mean is 1.0159 inches. 

A column of mercury 2 inches in height will counterbalance a pressure of .98 lbs. 
on a square inch. 

The practical estimate of the velocity of steam, when flowing into a vacuum, is 
about 1400 feet in a second when at an expansive power equal to the atmosphere ; 
and when at 20 atmospheres, the velocity is increased but to 1600 feet. 

And when flowing into the air under a similar power, about 650 feet per second, 
increasing to 1600 feet for a pressure of 20 atmospheres. 

The elasticity of the vapour of spirit of wine, at all temperatures, is equal to 
2.125 times that of steam. 



214. 



STEAM. 



STEAM ACTING EXPANSIVELY. 

To find the Mean Pressure of the Steam on the Piston. 

Rule.— Divide the length of the stroke, added to the clearance in- 
the cylinder at one end, by the length of the stroke at which the 
steam is cut off, added to the clearance, and the quotient will ex- 
press the relative expansion it undergoes. 

Find in the following table, in the column of expansion, a number 
corresponding to this ; take out the multiplier opposite to it, and 
multiply it into the full pressure of the steam per square inch as it 
enters the cylinder. 



Expansion. 



Table showing the Mean Pressure of Steam 

Multiplier. Expansion, i Multiplier. Expansion. 



I.O 


1.000 


3.4 


.654 


5.8 


.479 


1.1 


.995 


3.5 


.644 


5.9 


.474 


1.2 


.985 


3.6 


.634 


6. 


.470 . 


1.3 


.971 


3.7 


.624 


6.1 


.466 


1.4 


.955 


3.8 


.615 


6.2 


.462 


1.5 


.937 


3.9 


.605 


6.3 


.458 


1.6 


.919 


4. 


.597 


6.4 


.454 


1.7 


.900 


4.1 


.588 


6.5 


.450 


1.8 


.882 


4.2 


.580 


6.6 


.446 


1.9 


.864 


4.3 


.572 


6.7 


.442 


2. 


.847 


4.4 


.564 


6.8 


.438 


2.1 


.830 


4.5 


.556 


6.9 


.434 


2.2 


.813 


4.6 


.549 


7. 


.430 


2.3 


.797 


4.7 


.542 


7.1 


.427 


2.4 


.781 


4.8 


.535 


7.2 


.423 


2.5 


.766 


4.9 


.528 


7.3 


.420 


2.6 


.752 


5. 


.522 


7.4 


.417 


2.7 


.738 


5.1 


!516 


7.5 


.414 


2.8 


.725 


5.2 


.510 


7.6 


.411 


2.9 


.712 


5.3 


.504 


7.7 


.408 


3. 


.700 


5.4 


.499 


7.8 


.405 


3.1 


.688 


5.5 


.494 


7.9 


.402 


3.2 


.676 


5.6 


.489 


8. 


.399 


3.3 


.665 


5.7 


.484 







Multiplier* 



Example.— Suppose the steam to enter the cylinder at a pressure 
of 20 lbs. per square inch, and to be cut off at :i the length of the 
stroke of the piston. The stroke being 8 feet, 

8 feet = 96 inches + 1 for clearance := 97, 
i = 24 inches -|- 1 *' ■= 25. 

Then 97-^25 = 3.88, the relative expansion which falls between 
3.8 and 3.9. Referring to the table, the multiplier for 3.8 is .615, 
and the difference between that and the multiplier for 3.9 is .010. 
Hence, multiplying .010 by .8, and subtracting the product .008 from 
.615, the remainder, .607, is the multiplier for 3.88. Therefore, .607 
.X20 lbs. = 12 140 lbs. ix?r square inch, the mean effective pressure 
of the piston required. 

Specific gravity of steam at the pressure of the atmosphere .490, 
air being 1. 



STEAM. 215 

FOR WARMING APARTMENTS. 

Every cubic foot of water evaporated in a boiler at the pressure of the atmo- 
sphere will heat 2000 feet of enclosed air to an average temperature of 750,Tnd 
each square foot of surface of steam-pipe will warm 200 cubic feet of space. 

The force of steam is the same at the boiling point for every fluid. 

LOSS BY RADIATION. 

To ascertain the Loss of Heat per Square Foot in a Second. 

"""' T = feTerS oltK^'^^ ^^' ''' ^ '^'^ ^^^ ^^^^ '' '^^ «^--' 
I = length of the pipe in feet, 
d = diameter in inches, 
V = velocity in feet per second, 
R = radiation in degrees of heat. 

l.7l(T-t)__ ^ 

^» * Tredgold, 



216 STEAM-ENGINE. 



STEAM-ENGINE. 

It is not consistent with the plan of this work to enter fully into 
details of the steam-engine, and this article will be confined exclu- 
sively to some practical rules, the utility of which have been tested 
and their use adopted. 



CONDENSING ENGINES. 

Cylinder. The thickness of the metal is found by the following 
formula : 

— — ^x -i-T = thickness in inches, P representing pres- 

10000 d—2.5^'' ' ^ 

sure of steam in lbs., and d diameter of cylinder. 

For cylinders over 30 inches diameter, divide by 9000 ; over 40 
inches, by 8000 ; over 50 inches, by 6000 ; and over 60 inches, by 
5000. 

Condenser, The capacity of it should be i that of the cylinder. 

Air-pump. The capacity of it should be -J that of the cylinder. 

Steam and Exhaust Valves. Their diameter should give an area 
of 10 square inches for every 10000 cubic inches contained in the 
cylinder, and should lift J their diameter. 

Foot and Delivery Valves. Their dimensions should give an area 
of-^Q that of the airpump. 

Force Pumps. Their capacity 'should be yl^ to yi^ that of the 
cylinder. 

Injection Cocks. Their area should be sufficient to supply 70 times 
the quantity of water evaporated when the engine is working at its 
maximum, and in marine engines there should be three of them to > 
each condenser, viz., a Side, Bottom, and Bilge. 

The Side injection should have yV of an inch diameter of pipe forr 
every inch diameter of cylinder, the Bottom injection should have 
-j^* of an inch diameter of pipe for every inch diameter of cyhnder ; ; 
the Bilge injection is usually a branch of the Bottom injection pipe, , 
and may be of less capacity. 

Piston Rod. Its diameter should be ^ that of the cylinder. 

Beam. Its length from centres should be twice the stroke of the 
piston, and its depth -^ of its length. The strap at its smallest di- 
mensions should have at least y^^ the area of the piston rod, and its 
depth equal half of its breadth. 

* The proportion here given will admit of a sufficient quantity of water when the- 
engine is in operation in the Gulf Stream, where the water is at times at the tem- 
perature of 84°, and the quantity of water (wlien the steam is at 10 lbs. pressure) 
required to give it and the water of condensation a temperature of 100^, is 70 times 
^hat of the quantity evaporatetl 



STEAM-ENGINE. 217 

Beam Centres, The end centres should have each one, and the 
mam centre two and a half times the area of the piston rod 

The proportion for the strap, is when the depth of the beam is 
J that of the stroke ; consequently, when the depth is less, the area 
must be increased. 

Connecting Rods. Their diameter in the neck should be the same 
as that of the piston rod. The diameter of the centre of the bodv 
IS found in the following manner ; 

As .75 the stroke of the piston is to the length of the body of the 
rod so IS the area of the neck to the area of the centre of the body 

\\ hen two rods are used, each diameter should be JL that of the 
piston rod. ^ ^ 

u. ^^ti ■ ^°^'\ = '^^^" "''''* *''°"'<i •'« 7 that of the strap or 
head, their length 5 times their width, and their area I that of the 
rod, 3 

.hmlTX ?^.'Tfl^ ^'^'' ^'-^ ^^^'' ^^^^ ^^ ^h^i^ le^st section 
should be J that of the piston rod. 

^^Crank Pins. Their area should be H times that of the piston 

Cranks. When of Cast Iron, the dimensions of their Hub should 
be, in diameter twice that of the shaft upon which they are to be 
placed, and in depth i their diameter. 

The Small end should have its diameter -twice, and its depth once 
the diameter of the pin. ^ 

cv^^^^r of IFrozi^^Aj Iron, the diameter of the hub, compared to the 
shaft, should be as 8 to 4.5. The same proportion for the smaU end 
compared with the pin. ' 

Water Wheels, or Fly Shafts. See Rule, page 168. 



BOILERS. 

For every cubic foot capacity in the cylinder, when the length of 
the flues do not exceed 40 feet, they should have from 18 to 20 
square feet of fire and flue surface. 

There should be at least 10 times the space in the steam room 
that there is in the cylinder. 

Grates. For Wood, their area should be 4, and for Coal i the 
number of cubic feet in the cylinder. ^ 



T 



218 STEAM-ENGINE. 

NON-CONDENSING, OR HIGH-PRESSURE ENGINES. 

Cylinder. The thickness is found by the same rule that is ap- 
plied for that of a condensing engine. 

Steam Valves. Their area should be the same as given for con-- 
densing engines. 

Piston Rod. The diameter should be ^ that of the cylinder. 

Connecting Rods, Crank Pins, Straps, Cranks, and End and Main 
Centres, should bear the same proportion to the piston rod as in con- 
densing engines. 

Gibs and Keys. The same as in condensing engines. 

Force Pumps. Their capacity, when the pressure of the steam is 
not to exceed 60 lbs., should be Jj the contents of the cylinder; 
when not to exceed 130 lbs., ^, and in a similar ratio for higher 
pressures. 

Water Wheel, or Fly Shafts. See Rule, page 158. 



BOILERS. 

With plain cylindrical boilers without flues, there? should be 75 
square feet of fire and flue surface for every cubic foot capacity in 
the cylinder, when their length does not exceed 25 feet. 

With boilers having flues there should be 125 square feet of fire 
and flue surface when their length does not exceed 25 feet. 

Locomotive Boilers should have. 210 square feet of fire and flue 
surface for every cubic foot capacity in the cylinder. 

These proportions are for obtaining a pressure of 60 lbs. to the 
square inch. 

When of greater length, a corresponding increase of fire surface 
will be required. 

Grates. One square foot is suflicient for a horse's power. 



STEAM-ENGINE. . 219 



GENERAL RULES. 

Journals. Their length should exceed the diameter not less than 
in the proportion of 10 to 9, and in some cases the proportion can 
be increased in the ratio of 3 to 2 with advantage. 

Steam Pipes. Their area should exceed that of the steam valve. 

Front Links, i the length of the stroke, and ^ the diameter of 
the piston rod. 

Beams. To ascertain the vibration of their end centres at right 
angles to the plane of the cylinder, let L represent length of beam, 
and S stroke of piston. 

v^(L-^2)2—(S-H2)^—-=: vibration at each end. 
z 

Cast Iron Beams should always have, when of uniform thickness, 
their thickness yg- of their depth. 

Piston Rods of different materials should have their diameters in 
the following ratios : 

Cast iron 8 

Wrought iron 5 

Tempered steel 4 

Safety Valves * 10 inches area of valve for every 250 square feet 
fire surface. 



OF SATURATION IN MARINE BOILERS. 

100 parts of sea water contains 3 parts of its weight in saline 
matter, and is saturated when it contains 36 parts ; then, if the 
quantity in the boiler be taken as 100 parts of water, and 5 parts 
be used for steam, h parts blowed out ; to fix on the degree of satu- 
ration to contain x parts of saline matter, the quantity of salt en- 
tering and the quantity leaving in the same time, will be equal 

3^ 

when Z{s-\-h) = xh ; hence h z=z . 

x—3 
If X = 30, the water in the boiler will not reach to a higher degree 
of saturation when ^ of the quantity used for steam is allowed to 
escape. And as it requires but about } of the quantity of fuel to 
boil water that is required to convert it into steam, the loss of fuel 
will be ^X^ — ^V PSirt.^Tredgold. 



* Tredgold gives the following rule : Divide the area of the fire surface by the 
lbs. pressure (per steam gauge) of the steam, and the quotient will be the square 
of the diameter of the valve in inches. 



220 STEAM-ENGINE. 

SMOKE PIPES, OR CHIMNEYS. 

Their area at the base should always exceed that of the flue or 
flues. When wood is used, the area is required to be greater than 
for coal. 

The intensity of the draught is as the square root of the height. 



BELTS. 

Two 15 inch belts over a driver of six feet in diameter, running 
with a velocity of 2128 feet in a minute, transmit the power from 
the water-wheel at Rocky Glen Factory, the dimensions of which 
are given in page 179. 

An 11 inch belt over a driver of 4 feet in diameter, running from 
1200 to 2100 feet in a minute, will transmit the power from two 6 
inch cylinders having 11 inches stroke, and averaging 125 revolu- 
tions per minute, with a pressure of 60 lbs. per square inch. 

Two 6 inch belts over a driver 'of 5.9 feet in diameter, running 
2700 feet in a minute, will transmit the power from two 9 inch cyl- 
inders having 8 inches stroke, and averaging 150 revolutions per 
minute, with a pressure of 60 lbs. per square inch. 



STEAM-ENGINE. 221 

To find the Power of a Condensing Engine. 

Let *2? represent vacuum upon cylinder piston in lbs., 
S velocity of cylinder piston in feet per minute, 
n velocity of air-pump piston in feet per minute, 
*P mean effective pressure upon cylinder piston in lbs., 
m pressure upon cylinder piston necessary to overcome the 

friction of the air-pump and its gearing, 
h the lbs. pressure upon the air-pump piston, 
/ the lbs. pressure upon the piston necessary to overcome 

the friction of the engine. 
Where an Indicator is not used, estimate the value of i? at 9.5 lbs. 

The value of m is about 2 lbs. per square inch, that of ^, = r in 
pressure, and / is ^ of the pressure per steam gauge. 

Then ^^ ^- — -~^ = horses' power. 

Example. — The diameter of a cylinder is 60 inches, the stroke of 
the piston 10 feet, the revolutions 20 per minute, the diameter of the 
air-pump 46 inches, and the stroke 4 feet ; the pressure of the steam 
20 lbs. per square inch, cut off at \ the length of the stroke. 
Then v = area of 60x9.5 = 26860.3, 

8 = 10X2X20 =400, 

n = 4X2X20 =160, 

P =(per rule, page 214) 12.1X2827.4 = 34211.54, 
m =2827.4X2 =5654.8, 

b =area of 46x9.5 = 15788, 

/ =20X.2X2827.4 = 11309.6. 

34211.54 + 26860.3 — 11309.6 + 5654T8 X 400 — 160 X 15788 
33000 
526.984 horses' power. 

To find the Power of a Non-condensing Engine, 

SxP=7 ^ 

-3300r = ^'''"' P"^"'' 
f=^\ of the pressure per steam gauge. 

Example 1. — What is the power of an engine, the diameter of the 
cylinder being 10 inches, the stroke 4 feet, the pressure of the steeim, 
per gauge, 60 lbs., making 45 revolutions'? 

360 X 60— 7.5 X 78.54-^33000 = 44.982, Ans. 

2. The same with 30 lbs., cut off at J the stroke, and making 25 
revolutions 1 

200x21.25-3.75X78.54-^33000 = 8.02, Ans. 

* These vahies are best obtained by an Indicator. 
T2 



222 STEAM-ENGINE. 

The usual rule for either engine is, multiply the effective pressure 
upon the piston in lbs. per square inch by the velocity of the piston 
in feet per minute, and divide by 33000. 

To find the Volume the Steam of a Cubic Foot of Water occupies 
{separated from the Water) ^ the Elastic Force and Tempera- 
ture being given. 

Rule.— To the temperature in degrees add 459, multiply the sum 
by 38, and divide the product by the pressure in lbs. per square inch. 

Example.— The temperature is 291.5°, and the pressure 4 atmo- 
spheres, or 120 inches of mercury ; what is the volume 1 

An atmosphere is == 14.7 lbs. per square inch, and 14.7x4 = 

58.8 lbs. 

And 2 inches mercury = .98 lbs. per square inch ; therefore, 58.8 
-i-.98 = 60X2 = 120 inches. 

Then 295. lo_f-459x38-^58.8 =485.016 cubic feet, Am. 

What quantity of water will an engine of 10 inches cylinder, 4 
feet stroke, and making 45 revolutions per minute, require per hour 
at the pressure above given 1 

Area of 10 inches =78.54x48x2x45-M728 = 196.3 cubic feet 
of steam per minute. 

Then, as 485.016 : 1 : : 196.3 : .4047, and .4047x60 minutes = 
24.28 cubic feet of water per hour at a pressure of 58.8 — 14.7 = 44.1 
lbs. steam gauge. 

If it were required for 58.8 lbs. per steam gauge, the quantity 
would be in the proportion of their densities, viz., as .00210 to .00258 
(see table, page 212), or 54.18 cubic feet, independent of the quan- 
tity lost by waste, and the clearance of the piston in the cyhnder. 



To find the Power of an Engine necessary to raise Water to any 
Given Height. 

Rule.— Multiply the weight of the column by the velocity in feet 
per minute, and divide by 33000. 

Example. — It is required to raise a column of fresh water, 1& 
inches in diameter by 86 feet high, with a velocity of 128 feet per 
minute ; what power is necessary \ 

86 feet —2.31 feet, the height equal to 1 lb. per square inch = 
37.2 lbs. Area of 16 inches =201.x37.2 lbs. X 128 = 95708 1.6H- 
33000 = 29, horses' power. To which must be added an allowance 
for friction and waste, say \. 



STEAJVI-ENGINE. 223 

To find the Velocity necessary to Discharge a Given Quantity 
of Water in any Given Time. 

Rule. — ^Multiply the number of cubic feet by 144, and divide the 
product by the area of the pipe or opening. 

Example.— The diameter of the pipe is 16 inches, and the quanti- 
ty of water 179 cubic feet ; what is the velocity 1 
179Xl44-r201 = 128.2 feet, An^. 

To find the Area, the Velocity and Quantity being given. 
Rule.— Proceed as above, and divide the product by tlie velocity. 



224. 



COMBUSTION. 



COMBUSTION. 

Combustion is one of the many sources of heat, and denotes the combination of 
a body with any of the substances termed Supporters of Combustion : with refer- 
ence to the generation of steam, we are restricted to but one of these combinations, 
and that is Oxygen. 

All bodies, when intensely heated, become luminous. When this heat is produ- 
ced by combination with oxygen, they are said to be ignited ; and when the body 
heated is in a gaseous state, it forms what is called Flame. 

No bodies appear visible, even in a faint light, below about 870<^. 

Carbon exists in nearly a pure state in charcoal and in soot. It combmes with 
no more than 2§ of its weight of oxygen. In its combustion, 1 lb. of it produces 
sufficient heat to increase the temperature of 13000 lbs. of water l^. 

Hydrogen exists in a gaseous state, and combines with 8 times its weight of oxy- 
gen, and 1 lb. of it, in burning, raises the heat of 42000 lbs. of water lo. 

FUEL. 

With equal weights, that which contains most hydrogen ought, in its combus- 
tion, to produce the greatest quantity of heat where each kind is exposed under 
the most advantageous circumstances. Thus, pine wood is preferable to hard 
wood, and bituminous to anthracite coal. 

When wood is employed as a fuel, it ought to be as dry as possible. To produce 
the greatest quantity of heat, it should be dried by the direct application of heat. 
As usually employed, it has about 25 per cent, of water mechanically combined, 
the heat necessary for the evaporating of which is lost. 

Weight of sundry Fuels to form a Cubic Foot of Water at 52° into 
Steam at 220°. 



Newcastle coal . 
Pine wood (dry) . 
Oak wood (dry) . 



20.2 
12. 



Peat . 
Olive oil 
Coke . 



Lbs. 
30.5 

5.9 

9. 



Table showing the Heating Power of different Substances. 










Weight of wa- 








Weight of water 


ter converted 




Composition of combustible 


in lbs., heated 1° 


into steam by 1 


Name. 


portion. 




by 1 lb. of the 
combustible. 


lb. of combusti- 
ble, from 52® 
to 220=". 


defiant Gas . 


^ Hydrogen 
t Carbon . 


1 

■ 1 


12300 




Alcohol 


Hydrogen 


.1224 


11000 




(Spec. grav. .812) 


Carbon . 


.4785 






Olive Oil 


{ Hydrogen 
( Carbon . 


.133 

.772 


14500 


12. 


Beeswax, yellow . 


\ Hydrogen 
( Carbon . 


.1137 
.8069 


14000 


11. 


Tallow .... 






15000 


12. 


Oak wood, seasoned . 


{ Hydrogen 
( Carbon . 


.057 
.525 


4600 


3.90 


, dried on a 










stove .... 






5960 


5.12 


^ allowing 20 










per cent, loss 






5660 


4.85 


Pine, seasoned 






5466 


4.66 


Coal, Newcastle . 


j Hydrogen 
( Carbon . 


. .0416 
. .7516 


9230 


7.90 


, Welsh . 






11840 


10.1 


, Anthracite . 


Carbon . 


. .88744 


9560 


8. 


, Cannel . 


{ Hydrogen 
\ Carbon . 


. .0393 
. .722 


9000 


7.7 


Coke .... 
Peat .... 


Carbon . 


. .84 


9110 
3250 


7.7 



COMBUSTION. 



225 



Small coal produces about ^ the effect of good coal of the same species. 
The averages of the above, for practical results, may be set down as follows : 



Oak 
Piiie 
Coke 



Heated 1= by lib. 
. 4500 lbs. 
. 5000 " 
. 8600 " 



Bituminous coal 
Anthracite 



Hea'ed 1° by 1 lb. 

. 9200 lbs. 
. 7800 " 



Different fuels require different quantities of oxygen ; for the different kinds of 
coal, it varies from 1.87 to 3 lbs. for each lb. of coal. 60 cubic feet of air is neces- 
sar>' to afford 1 lb. of oxygen ; and making a due allowance for loss, nearly 90 cubic 
feet of air will be required in the furnace of a boiler for each lb. of oxygen. 

The quantity of air and smoke for one cubic foot of water converted into steam 
at 220O is, for coal about 2000, and for hard wood about 4000 cubic feet. 

Table showing the Results of Mr. BulVs Experiments upon Wood. 



Woods. 


Weight of 
a cord. 


Compara- 
tive value 
per rord. 


Woods. 


Weight of 
a cord. 


Compara- 
tive value 
per cord. 


Shell -bark Hickory 
Pig-nut Hickory . 
Red-heart Hickory 
White Oak . . . 
Red Oak . . . . 


Lbs. . 
4469 
4241 
3705 
3821 
3254 


100 
95 
81 
81 
69 


Hard Maple . . . 
Jersey Pine . . . 
Yellow Pine . . . 
White Pine . . . 


Lbs. 
2878 
2137 
1904 
1868 


60 
54 
43 
42 



Pounds of Ice melted by the following Fuels : 



Good coal 
Coke . 
Charcoal 



90 
94 
95 



Wood (hard) 
Peat . 
Hydrogen gas 



92 

19 

370 



When bituminous coal is subjected to destructive distillation, about § of its 
weight is left, in the form of coke. 



Kelative Value oj 
Seasoned oak 


f the pit 
125 


owing riuls by Weight: 
Charcoal . . . . 


285 


" " artificially 


140 


Peat 


115 


Hickory 


137 


Welsh coal . 


312 


White pine . 


137 


Newcastle *' . . . 


309 


Yellow pine 


145 


Belgium " . . . 


316 


Good coke . 


285 


Anthracite, French 


290 


Inferior *' . . . 


222 


" Pennsylvamia . 


250 



ANALYSIS OF FUELS. 



Carbon . 
Hydrogen 
Nitrogen . 
Oxygen . 



Volatile matter 

Charcoal 

Ashes 



fewcastle Coal, 

cakin? kind. 

75.28 

4.18 
15.96 
4.58 


Cannel Coal. 

64.72 

21.56 

13.72 

0.00 

100. 

Ash. Maple. 
81.3 79.3 
17.9 20. 

.7 .7 


Cumberland Coal, 
American. 
80. 
Bitumen, 18.40 
Ash, 1.60 


100. 

Oak. 

76.9 

22.7 
.4 


100. 

Chestnut. Norway Pine. 
76.3 80.4 
23.3 19.2 
.4 .4 



An increase in the rapidity of combustion is accompanied by a diminution in the 
evaporative efficiency of the combustible. 



226 COMBUSTION. 

ANTHRACITE COAL. 

The results of late and accurate observations upon the burning 
of anthracite coal, with the aid of a blast, gives an expenditure of 5 
lbs. per horse power per hour. 

The best anthracites contain about 95 per cent, of inflammable matter, principally 
carbon. 

1.84 tons coal are required for the smelting and heating of the blast to make 1 ton 
pig iron. 

578304 cubic feet of air are required for the blast to make 1 ton of iron. 

CHARCOAL. 

The best quality is made from oak, maple, beech, and chestnut. 

Wood will furnish, when properly burned, about 16 per cent, of coal. 

A bushel of coal from hard wood weighs about 30 Ifts., and from pine 29 lbs. 

COKE. 

Sixty* bushels Newcastle coals (lumps) will make 92 bushels good coke, and 60 
bushel's (fine) will make 85 bushels of a similar quality. 

60 bushels Newcastle and Picton coal (one half of each) makes 84 bushels infe- 
rior ; 60 bushels Picton, or Virginia coal, makes 75 bushels of bad. 

A bushel of the best coke weighs 32 lbs. 

Coal furnishes 60 to 70 per cent, of coke by weight. 

1 lb. in a common locomotive boiler will evaporate 7^ lbs. water at 212° into 



MISCELLANEOUS. 

One pound of anthracite coal in a cupola furnace will melt 5 lbs. of cast iron ; 80 
bushels bituminous coal in an air furnace will melt 10 tons cast iron. 

When one bushel bituminous coal per hour will produce steam of the expansive 
force of 15 lbs. per square inch, 1* bushels will give 50 lbs., and 2 bushels 120 lbs. 

One lb. of Newcastle coal converts 7 lbs. of boiling water into steam, and the 
time, 6 times that necessary to raise it from the freezing to the boiling point. And a 
bushel will convert 10 cubic feet of water into steam. 

* Winchester bushel = 2150.42 cubic inches. 



WATER. 



227 



WATER. 

Fresh Water. The constitution of it by weight and measure is, 

By weight. By measure. 

Oxygen 88.9 1 

Hydrogen 11.1 2 

One cubic inch at 62°, the barometer at 30 inches, weighs 252.458 
grains, and it is 830.1 times heavier than atmospheric air. 

A cubic foot weighs 1000 ounces, or 62 i lbs. avoirdupois ; a col- 
umn 1 inch square and 1 foot high weighs .434028 lbs. 

It expands i of its bulk in freezing, and averages .0002517, or ^-X^ 
for every degree of heat from 40° to 212°. Maximum density, 39.38°. 



Table of Expansion at different Temperatures. 



Temperature. 


Expansion. 


Temperature. 


Expansion. 


12° 
22° 
32° 
40° 


.00236 
.00090 
.00022 
.00000 


64^ 
102° 
212° 


.00159 
.00791 
.04330 



Showing an increase in bulk from 40° t-o 212° of 5^^, equal to 1 
cubic foot in every 23.09 feet. 

The height of a column of ( ] ^\- P^^ ^9^^^^ ^n^^' ^^ 2.31 feet, 
water at 60°, equivalent to the ( ] ' circular " - 2.94 - 
pressure of ) 1 mch of mercury, "1.133*^ 

\ the atmosphere . *' 34. 

River or canal water contains ^ ^ of its volume of gaseous mat- 

Spring or well water " __i_ i ter. 

A cubic inch weighs .03611 of a lb., and at 212° has a force of 
29.56 inches mercury. 

Sea Water, according to the analysis of Dr. Murray, at the spe- 
cific gravity of 1.029, contains. 
Muriate of soda 
Sulphate of soda 
Muriate of magnesia 



Muriate of lime 



220.01 = -1^ 
33.16=^ 

1 

23? 



"2T^ 

303.09=: 3L 



42.08 : 

7.84 = - 



Table showing the Deposites that take place at different Degrees of 
Saturation and Temperature. 



When 1000 parts were reduced by evaporation. 



Quantity of sea water. 



1000 
299 
102 



Boiling point. 



214° 
217° 

228° 



Salt in 100 parts. 



3. 

10. 
29.5 



Nature of deposi te. 



None. 

Sulphate of lime. 

Common salt. 



228 



WATER* 



Boiling Point at different Degrees of Saturation. 



Proportion of salt in 100 parts by 
weight. 



Saturated ) 
solution 5 



36.37 

33.34 
30.30 

27.28 
24.25 
21.22 



Boiling point. 



226.° 

224.9° 
223.7^ 

222-5° 
221.4° 
220.2° 



Proportion of salt in 
100 pans by weight. 



18.18 

15.15 

12.12 

9.09 

6.06 

3.03 

Sea water 



\ 



Boiling point. 



219.0 

217.9° 

216.7° 

215.5° 

214.4° 

213.2° 



Salt Water. A cubic foot of it weighs 64.3 lbs. ; a cubic inch, 



.03721 lbs. 

The height of a column of j 
water at 60°, equivalent to the 
pressure of . 

(Specific gravity, 1029). I 



1 lb. per square inch, is 2.37 feet, 
1 " " circular " " 3.02 " 
1 inch of mercury, " 1.165" 
the atmosphere . ''34.98 " 



MOTION OF BODIES IN FLUIDS. 



229 



MOTION OF BODIES IN FLUIDS. 



Table of the Weights required to give different Velocities to several 
different Figures. 

The diameter of all the figures but the small hemisphere is 6.375 
inches, and the altitude of the cone 6.625 inches. 

The small hemisphere is 4.75 inches. 

The angle of the side of the coae and its axis is, consequently, 
25° 42' nearly. 



Velocity 


Cone. 


Whole 
globe. 


Cylinder. 


Hennisphere. 


Small hem- 


per second. 


Vertex. 


Base. 


Flat. 


Round. 


isphere. 


feet. 


oz. 


oz. 


oz. 


oz. 


oz. 


oz. 


oz. 


3 


.028 


.064 


.027 


.050 


.051 


.020 


.028 


4 


.048 


.109 


.047 


.090 


.096 


.039 


.048 


5 


.071 


.162 


.068 


.143 


.148 


.063 


.072 


6 


.098 


*.225 


.094 


.205 


.211 


.092 


.103 


7 


.129 


.298 


.125 


.278 


.284 


.123 


.141 


8 


.168 


.382 


.162 


.360 


.368 


.160 


.184 


9 


.211 


.478 


.205 


.456 


.464 


.199 


.233 


10 


.260 


.587 


.255 


.565 


.573 


.242 


.287 


12 


.376 


.850 


.370 


.826 


.836 


.347 


.418 


15 


.589 


1.346 


.581 


1.327 


1.336 


.552 


.661 


16 


.675 


1.546 


.663 


1.526 


1.538 


.634 


.754 


20 


1.069 


2.540 


1.057 


2.528 


2.542 


1.033 


1.196 


Propor. 
number 


126 


291 


124 


285 


288 


119 


140 



From this table several practical inferences may be drawn. 

1. That the resistance is nearly as the surface, the resistance in- 
creasing but a very little above that proportion in the greater sur- 
faces. 

2. The resistance to the same surface is nearly as the square of 
the velocity, but gradually increasing more and more above that 
proportion as the velocity increases. 

3. When the hinder parts of bodies are of different forms, the re- 
sistances are different, though the fore parts be alike. 

4. The resistance on the base of the hemisphere is to that on the 
convex side nearly as 2.4 to 1, instead of 2 to 1, as the theory as- 
signs the proportion. 

5. The resistance on the base* of the cone is to that on the vertex 
nearly as 2.3 to 1. And in the same ratio is radius to the sine of 
the angle of the inclination of the side of the cone to its path or axis. 
So that, in this instance, the resistance is directly as the sine of the 

* This is a complete refutation of the popular assertion, that a taper spar will 
tow in water easiest when the base is foremost. 

u 



230 MOTION OF BODIES IN FLUIDS. 

angle of incidence, the transverse section being the same, instead 
of the square of the sine. 

6. Hence we can find the altitude of a column of air, the pressure 
of which shall be equal to the resistance of a body moving through 
it with any velocity. 

Thus, let a = the area of the section of the body, similar to any of tliose in the 
table, perpendicular to the direction of motion, 
R = the resistance to the velocity, in the table, and 

X = the altitude sought, of a column of air whose base is a and its pressure R. 
Then ax = the contents of the columns in feet, and 1.2 ax, or ^ ax its weight in 
ounces. 

R 
Therefore, 6 ^ x = R, and a; = #X~ is the altitude sought in feet, namely, 5 of the 
^ ° a 6 

quotient of the resistance of any body divided by its transverse section, which is a 
constant quantity for all similar bodies, however different in magnitude, since the 
resistance R is as the section a, as by article 1. 

When a = |- of a foot, as in all the figures in the foregoing table except the small 

R It; 

hemisphere, then x = | X — , becomes x = — R, where R is the resistance in the 

Q a ^ 

table, to the similar body. 

If, for example, we take the convex side of the large hemisphere, whose resist- 
ance is .634, or at a velocity of 16 feet per second, then R = .634, and x = — R = 
2.3775 feet, is the altitude of the column of air whose pressure is equal to the re- 
sistance on a spherical surface, with a velocity of 16 feet. 

And to compare the above altitude with that which is due to the given velocity, 
it will be 322 ^iqz. . iq . 4^ t^e altitude due to the velocity 16, which is near double 
the altitude that is equal the pressure. And as the altitude is proportional to the 
square of the velocity^ therefore, in small velocities the resistance to any spheri- 
cal surface is equal to the pressure of a column of air on its great circle, whose al- 
titude is ^, or .594 of the altitude due to its velocity. 

But if the cj^linder be taken, where resistance R = 1.526, then x=: -^Rr=5.72, 
which exceeds the height 4, due to the velocity, in the ratio of 23 to 16 nearly. 
And the difference would be still greater if the body were larger, and also if the 
velocity were more. 

If any body move through a fluid at rest, or the fluid move against 
the body at rest, the force or resistance of the fluid against the body 
will be as the square of the velocity and the density of the fluid ; 
that is, R:=zdv^. 

For the force or resistance is as the quantity of matter or particles struck, and the 
velocity with which they are struck. But the quantity or number of particles 
struck in any time are as the velocity and the density of the fluid. Therefore, the 
resistance or force of the fluid is as the density and square of the velocity. 

The resistance to any plane is also more or less, as the plane is greater or less, 
and therefore the resistance on any plane is as the area of the plane a, the density 
of the medium, and the square of the velocity ; that is, R = adv^. 

If the motion be not perpendicular, but oblique to the plane, or to the face of the 
body, then the resistance in tlie direction of the motion will be diminished in the 



MOTION OF BODIES IN FLUIDS. 231 

triplicate ratio of radius to the sine of the angle of inclination of the plane to the 
direction of the motion, or as the cube of radius to the cube of the sine of that an- 
gle. So that R = adv^s^, 1 = radius, and s = sine of the angle of inclination. 

The real resistance to a plane, from a fluid acting in a direction perpendicular to 
its face, is equal to the weight of a column of the fluid, whose base is the plane 
and altitude equal to that which is due to the velocity of the motion, or through 
which a heavy body must fall to acquire that velocity. 

The resistance to a plane rimning through a fluid is the same as the force of the 
fluid in motion with the same velocity on the plane at rest. But the force of the 
fluid in motion is equal to the weight or pressure which generates that motion, 
and this is equal to the weight or pressiu-e of a column of the fluid, the base of 
which is the area of the plane, and its altitude that which is due to the velocity. 

1. If a be the area of a plane, v its velocity, n the density or specific gravity of 
the fluid, and i ^=: 16.0833 feet ; then, the altitude due to the velocity v being — , 
therefore aXnX^ = -g— , will be the whole resistance or force R. 

2. If the direction of motion be not perpendicular to the face of the plane, but 
ODhque to it, in an angle ; then R = . . 

3. If W represent the weight of the body, a being resisted by the absolute force 
R ; then the retarding force /, or 5 will be ^^!!^ 

The resistance to a sphere moving through a fluid, is but half the resistance to 
its great circle, or to the end of a cylinder of the same diameter, moving with an 
equal velocity. 

^ = ^^ > being the half of that of a cylinder of the same diameter, R repre- 
senting radius. 

Illustration.— A 9 lb. iron ball, the diameter being 4 inches, when projected 
at a velocity of 1600 feet per second, will meet a resistance which is equal to a 
weight of 132.66 lbs. over the pressure of the atmosphere. 



232 



AIE. 



AIR. 



100 Cubic Inches of atmospheric air, at the surface of the earth, 
when the barometer is at 30 inches, and at a temperature of 60^' 
weighs 30.5 grains, being 830.1 times lighter than water. ' 

Specific gravity compared with loaier, .001246. 

The atmosphere does not extend beyond 50 miles from the earth's 
surface. 

The mean weight of a column of air a foot square, and of an al- 
titude equal to the height of the atmosphere, is equal to 2116 8 lbs 
avoirdupois. 

It consists of oxygen 20, and nitrogen 80 parts ; and in 10 000 
parts there are 4.9 parts of carbonic acid gas. 

The mean pressure of the atmosphere is usually estimated at 14 7 
lbs. per square inch. 

13.29 cubic feet of air weigh a lb. avoirdupois, hence 1 ton of air 
wall occupy 29769.6 cubic feet. 

The rate of expansion of air, and all other Elastic Fluids, for all 
temperatures, is uniform. 

From 320 to 212^ they expand from 1000 to 1376, equal to -1-* 
of their bulk for every degree of heat. "* "^^ 

See Heat, page 201. 



'- 5-|g equal .002087 for each degree. 



DIMENSIONS AND WEIGHTS OF GUNS, SHOT, AND SHELLS. 233 



to < 

O S^« 

3 S^ S 
■= i" M 









s-§ i-5-§ 



C5 



Co 00 

c§ Cos" 



to CO CO »^ ^ I— • 
rf»- t3 t3 to to 00 O 



^ 3 



CD r^ 



<lK)tO W 



to to 



cn CJ 35<!<? 00 Ob 
6oifi.|!fi. ■ O* " §■ 



-<f 4^. Oi 05 

b* toto 



w cx<! tn o:> 

to O 00 CO cn 



CO 

to 



OJOO CTK? 

cn cn OOt— ' 
Ci * C5tO 



OO 

COO 



cn ►— ocn 



to 



tototototooto" 

CO Cn cn <X> <X) CO rf^g. 

COtOtO ' lifi." top 

cn cn 



CO ►f^ CO 4^ 
05 05 05 to 
<f t3 ^ 



►p>-c;i 

OS CO 



cn cn 05 05 <J -^ 05 OO . 

bo bo^rf:^' c#^' I 

to ts e- H-O ? 

CO 00 5 



§3o I 



^-^ tOH 

cn C 



5 — to.-*- 
ZXt cn or " 



^ ■ 



to 

to 

o 



00 00 <J 03 
GO OOO to 

O OCn O 



COt^ 

OCn 
OO 



i^Cn 

5OC0 
50»— 



00 00 cn IX) 
CO 05 00 Cnt^ 
00000» 
OOO OO" 



13 



cn 

CO 



CO hf^ hF^ cn 
cn cn cn 05 

00 to to 00 



CO rfs"- rfi^cn 



CO ^^ cn 

cn kP^ >P>- 05 
>f^ 05 05 »— ' 



►^cn 
Cn^ 
to<j 



cn 

C5 
00 



00 



5 05 05<J CO^ 

3 00 00 00 00 S rrS-S- 
" CO CO cn cn • " ■" 



rf^CJi 
05 00 



cn 

o 



5 05 05 <{0^ 

5 00 boo oi" 

J05 0500* 



tf^Cn 
050 



<i Cnc 

00 05^ 
O t-C 



5 05 05-<J O^ 

■" <j<jbobos 

3 05 05 OO' 



•-^t0O5S 
cn cn cn O' 



Thickness 
of shells. 



. 00 00 Oi 

COCOtO 

to 



MOO 
Cn<! 

050 



to 

CD 

to 



COCO 

coco 
bo 



►^ ►Pk 05 C0_ 
COCO O Ob- 
COCOCn ' 



p: ^ 



00 to 

cobi 



tf!^ .- t 

►^ 05 t 

05 to li 

to tOf 



3 CO CO ►P^ 00 

"-coco bco 

- 00 00 to CO 



I cn cn O 

I cococo 



to 

05 



' 7- ^ tOCn^ 

: 05 o>05i^ = 

3 >f^ 4^ 05 to 



►^ffi-oc;! 

H- 4^ to CO 

" * bo" 



to to 

00 cn 

oto 



t;^ 
<? 



3^ to 

5 CO OOO 05t- 

jH- pcocog- 
t— <{ boco <j * l^bo' 



J to tOH 

> — toe 

5*^<?H 



= o ^ 



^3 



O COOOrfa* 



k^OiOS-^OOCncnS Windage. 
CO H- 00 



Oi05 

^to 



I I 



05 
to 



05-<{, 



'3o3 



I I ^^ 

tO<! 

Cncn 



I I 



Mill 



Length of 
chamber. 



00 00 00 00 

o 0000 



0000 00 00 00 00 00 00 00 00 00 

>^tf»' >f^ tf^ »— j^ to 4i>- CO t^ rfii 

©o o o ;o©coo^oo 

U2 



234 WEIGHT AND DIMENSIONS OF LEADEN BALLS. 



WEIGHT AND DIMENSIONS OF LEADEN BALLS. 



Table showing the Number of Balls in a Pound, from l.-^ihs to 
TWF ^f ^^ ^^^^ Bore. 



Diam. 


Diam. 


Number 


Diam. 


Diam. 


Number 


Diam. 


Diam. 




in parts of 


in decimals 


of balls in a 


in parts of 


in decimals 


of balls in a 


in parts of 


in decimals 


of balls in a 


an inch. 


of an inch 


pound. 


an inch. 


of an inch. 


pound. 


an inch. 


of an inch. 


pound. 




1.670 


1 




.570 


25 




.301 


170 


*i-nr 


1.326 


2 




.537 


30 




.295 


180 




1.157 


3 




.510 


35 




.290 


190 




1.051 


4 


* 1 
2 


.505 


36 




.285 


200 


* H 


.977 


5 


.488 


40 




.281 


210 


.919 


6 




.469 


45 




.276 


220 


* i 


.873 


7 




.453 


50 




.272 


230 


.835 


8 


*Tt 


.426 


60 




.268 


240 


* T 


.802 


9 


.405 


70 




.265 


250 


.775 


10 




.395 • 


75 




.262 


260 


.750 


11 




.388 


80 




.259 


270 




.730 


12 


* i 


.375 


88 




.256 


280 




.710 


13 




.372 


90 


* i 


.252 


290 


*u 


.693 


14 




.359 


100 


.249 


300 


.677 


15 




.348 


110 




M7 


310 




.662 


16 




.338 


120 




.244 


320 




.650 


17 




.329 


130 




.242 


330 


* 5 


.637 


18 




.321 


140 




.239 


340 


.625 


19 




.314 


150 




.237 


350 




.615 


20 




.307 


160 









* The exact decimals would be as follows : 



1 5 

7 
H 



1.3125 
.9375 



.8750 



13 



1 



.8125 



.5000 



7 

TF 
5 

1 

4 



.4375 
.3125 
.2500 



Expansion of Shot heated to a White Heat. 



Expansion 



Inches. 
0.11 



Inches. 
0.10 



Inches. 
0.08 



Inches. 
0.06 



Inches. 
0.04 



Experiment at 
Fort Monroe, 1839. 



WEIGHT AND DIMENSIONS OF SHOT. 



GRAPE. 



CALIBRE OF 


8 Inch. 


42 


32 


24 


18 


12 


Diameter of high gauge . 
" low gauge . 


Inches. 
3.60 
3.54 


Inches. 
3.17 
3.13 


Inches. 
2.90 

2.86 


Inches. 

2.64 
2.60 


Inches, 
2.40 
2.36 


Inches. 
2.06 
2.02 


Mean weight in lbs. . . 


6.24 


4.25 


3.25 


2.45 


1.83 


1.19 



WEIGHT AND DIMENSIONS OF SHOT. 



235 



CANISTER. 





42 


32 


24 and 
8 inch 
how- 
iizer. 


18 


12 


9 and 
24 1b. 
how. 
ilzer. 


6 


12 lb. howitzer. 


Calibre of 


Field. 


Mountain 


Diam. of high gauge, 
" low gauge, 


Ins. 

2.26 

2.22 


Ins. 
2.06 
2.02 


Ins. 

1.87 
1.84 


Ins 
1.70 
1.67 


Ins. 
1.49 
1.46 


Ins. 

1.35 
1.32 


Ins. 
1.17 
1.14 


Ins. 
1.08 
1.05 


Ins. 

Musket 

ball. 


Mean weight in lbs., 


1.57 


1.19 


.90 


.67 


.45 


.33 


.235 


.17 


.056 



CARCASSES, 
lis inch. 1 10 inch.) 8 inch. I 42 i 32 t 24 i 18 I 12 



Mean weight in lbs. . . | 194 | 87.63 | 43.62 | 29.45 | 21.60 | 15.84 | 12.15 | 8 



BRONZE FOR CANNON. 
Copper 90, Tin 10. 

Specific gravity is greater than the mean of copper and tin, vix.* 
8.766. 



236 DIMENSIONS AND WEIGHTS OF GUNS, SHOT, AND SHELLS. 



ffi 

m 
o 

m 
m 

iz; 

P 
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K r; rf 


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c; rt 








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PENETRATION OF SHOT AND SHELLS. 



237 



PENETRATION OF SHOT AND SHELLS. 



^ PENETRATION IN MASONRY. 
Experiments at Fort Monroe Arsenal in 1839. 



Calibre. 


Charge. 


Elevation. 


Distance. 


Mean penetrat 


on. 




Dressed 
granite. 


Poioniac 
freestone. 


Hard 
briCK. 


Shot. 
32 Pounder (Gun) . 

Shell 
8 Inch Howitzer ) 

Seacoast ) 


Lbs. 
8 

6 


1° 

1° 35' 


Yards. 

880 
880 


Inches. 

3.5 
1. 


Inches. 

12. 
4.5 


Inches. 

15.25 
8.5 



The solid shot broke against the granite. 

The shells broke into small fragments against each of the three 
materials. 



PENETRATION IN WHITE OAK. 
Experiments at New- York Harbour in 1814. 



Calibre. 


Charge. 


Distance. 


Penetration. 


Remarks. 


32 Pounder . j 


Lbs. 
11 
11 


Yard?. 

100 
150 


Inches. 

60 
54 


Shot wrapped so as 
to destroy the wind- 
age. 



PENETRATION IN COMPACT EARTH 
(Half sand, half clay). 



Calibre. 


Charge. 




Distances in 


yards. 






27 


109 


32S 


1094 


Shot. 




Inches. 


Inches. 


Inches. 


Inches. 


6.885 


* 


109.1 


102.4 


93.4 


69.7 


Shells. 












8.782 


4.4 lbs. 


*48.4 


*45.3 


38.6 


23.2 


Musket 


154 grains 


9.85 


8.6 


4.3 





The penetrations in other kinds of earth are found by multiplying 
the above by 63 for sand mixed with gravel; by 0.87 for earth 
mixed vi^ith sand and gravel, weighing 125 lbs. to a cubic foot; by 
1.09 for compact mould and fresh earth mixed with sand, of half 
clay; by 1.44 for wet potter's clay; by 1.5 for light earth, settled- 
and by 1.9 for light earth, fresh. 



* With this charge, and at these distances, the shells were often broken. 



238 



PENETRATION OF SHELLS. 



PENETRATION OF SHELLS. 



Eleva- 


Distance 


In Compact Earth, 


In Oak Wood, 


In Masonry, 


tion. 




Sins. 


10 ins. 


12 ins. 


8 ins. 


10 ins. 


12 ins. 


S ins. 


10 ins. 


12 ins. 




Yards. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


laches. 


Inches. 


Inches. 


Inches. 


30<^| 


656 


7.8 


17.7 


19.6 


3.9 


7.8 


8.6 


1.9 


3.5 


3.9 


1312 


9.8 


25.6 


27.5 


4.7 


11.8 


13.7 


2.3 


4.7 


5.1 


45° "I 


656 


11.8 


19.6 


21.6 


5.9 


9.8 


10.6 


3.1 


3.9 


4.3 


1312 


15.7 


27.5 


29.5 


7.8 


13.7 


15.7 


3.9 


5.5 


5.9 


60oj 


656 


19.6 


29.5 


31.5 


8.6 


13. 


14.5 


4.3 


5.9 


6.3 


1312 


21.6 


31.5 


33.4 


9.8 


13.7 


15.7 


4.7 


6.3 


6.6 


Falling with ( 




















maximum < 


23.6 


33.4 


35.4 


9.8 


13.7 


15.7 


4.7 


6.6 


7. 


velocity. ( 





















The penetration in other kinds of earth and stone may be obtain- 
ed by using the coefficients given for the other tables. For woods, 
use for beech and ash 1, for elm 1.3, for white pine and birch 1.8, 
and for poplar 2. 

144 grains of powder in a musket, at 5 yards' distance, will pro- 
ject a ball 3 inches into seasoned wh*ite oak, and 100 grains in a 
rifle, at the same distance, 2.05 inches. 



MISCELLANEOUS. 



239 



MISCELLANEOUS. 



RECAPITULATION OF WEIGHTS OF VARIOUS SUBSTANCES. 











Cubic foot in pounds. 


Cubic inch in pounds. 


*Cast Iron .... 


450.55 


.2607 


t Wrought Iron . 








486.65 


.2816 


Steel 








489.8 


.2834 


tCopper . 








555. 


.32118 


Lead 








708.75 


.41015 


Brass 








537.75 


.3112 


Tin 








456. 


.263 


is White Pine . 








29.56 


.0171 


Salt Water (sea) 








64.3 


.03721 


Fresh Water . 








62.5 


.03611 


Air ... 




• 




.07529 




Steam . 








.0350 


— 



Weights of a Cubic Foot of various Substances in 
ordinary use. 



Loose earth or sand 


Lbs. 

. 95 


Clay and stones 


Lbt. 

. 160 


Common soil . 


. 124 


Cork 




15 


Strong soil 


. 127 


Tallow 




59 


Clay 


. 135 


Brick 




. 125 




SLATING. 








Sizes of Slates. 






Doubles . 




14 by 6 


inches, 


Ladies' 




15 '' 8 




Countess . 




22 *'ll 




Duchess . 




26 *-15 




Imperial and Patent 




32 "26 




Rags and Queens 




39 "27 





* From the West. Point Foundry Association at Cold Spring, N. Y. Other ex- 
periments have given .2613 as the weight of a cubic inch, 
t Ulster Iron Company, Saugerties. N. Y. 
t From Phelps, Dodge, & Co.'s Works, in Derby, Conn. 



240 



MISCELLANEOUS. 



CAPACITY OF CISTERNS IN U. S. GALLONS. 



For each 10 Inches m Depth. 



2 feet diameter 

2i 

3 

3^ 

4 

4i 

5 

5i 

6 

6i 

7 

7i 



19.5 

30.6 

44.06 

59.97 

78.33 

99.14 

122.40 

148.10 

176.25 

206.85 

239.88 

275.40 



8 


feet diameter . 313.33 


8i 




. 353.72 


9 




. 396.56 


9i 




. 461.40 


10 




. 489.20 


11 




. 592.40 


12 




. 705. 


13 




. 827.4 


14 




. 959.6 


15 




. 1101.6 


20 




. 1958.4 


25 




. 3059.9 



TABLE OF COMPOSITIONS BRASS, ETC. 



Copper. 


Tin. 


Zinc. 


2 





1 


3 





1 


4 




i 


6 







.5 




i 


8 







9 







3 





1 


10 


1 





78 


22 


5.6, and leaa 4.3 J 


80 


10 



For Yellow Brass. 
Spelter. 

Lathe brushes. 
Shaft bearings. 

(hard). 
Wheels, boxes, cocks, &c. 
Gun metal. 
Brass. 
Valves. 

Bells and Gongs. 



SIZES OF NUTS, EQUAL IN STRENGTH TO THEIR BOLTS. 



Diameter of bolt 


Short diameter 


Diameter of bolt 


Short diameter 


Diameter of bolt 


Short diameter 


in inches. 


of nut in inches. 


in inches. 


of nut in inche?. 


in inches. 


of nut in inches- 


1 


'3 


If 


2tV 


21 


4rt 


3 


5 


n 


2H 


2| 


4| 


^ 


7 


If 


2i 


2|- 


4il 


1 


ItV 


1^^ • 


3^ 


n 


H 


3 
4^ 


1^ 


li 


31 


3 


5| 


i 


lA 


2 


3^ 


3| 


5i 


1 


i| 


2| 


3| 


3i 


6/* 


u 


2 


21 


4 


sr 


6| 


u 


n 


21 


4| 


4 


n 



Note. — The depth of the head should equal the diameter of the bolt ; the den- 
of the nut should exceed it in the proportion of 9 or 10 to 8. 



MISCELLANEOUS. 



241 



SCREWS. 



Table showing the Number of Threads to an Inch in V thread Screws. 



Diam. in inches, 
No. of threads, 

Diam. in inches, 
No. of threads. 



20 

H 

6 



5 



7 



1 1 



18 16 14 12 11 10 9 



7 



Ij Ig- A& 2^ ^2 -^4 O o^- 3-2 

5 4J 4i 4 4 3i 3| 3^ 3^ 



^4 

2^ 



6 
2* 



Diam. in inches, 3| 4 4]^ ^ 4J 5 5j 

No. of threads, 3 3 2 J 2 J 2| 2| 2f 

The depth of the threads should be half their pitch. 

The diameter of a sc^ew, to work in the teeth of a wheel, should 
be such that the angle of the threads does not exceed 10°. 



Table of the Strength of Copper at different Temperatures, 



Temperature. 


strength in lbs. 


Temperature. 


strength in lbs. 


Temperature. 


strength in lbs. 


122° 


33079 


482° 


26981 


801° 


18854 


212^ 


32187 


545° 


25420 


912° 


14789 


302° 


30872 


602° 


22302 


1016° 


11054 


392° 


27154 











Franklin Institute. 



DIGGING. 

23 cubic feet of sand, or 18 cubic feet of earth, or 17 cubic feet 
of clay, make a ton. 

18 cubic feet of gravel or earth before digging, make 27 cubic feet 
when dug. 



COAL GAS. 

A chaldron of bituminous coal yields about 10.000 cubic feet of gas. 
Gas pipes i inch in diameter supply a light equal to 20 candles. 
1.43 cubic feet of gas per hour give a light equal to one good 
candle. 

1.96 cubic feet equal four candles. 
3. '* " '« ten 

X 



242 



MISCELLANEOUS. 



ALCOHOL 
Is obtained by distillation from fermented liquors. 

Proportion of Alcohol in 100 parts of the following Liquors : 



Scotch Whiskey 


54.32 


Sherry 




19.17 


Irish " . 


53.9 


Claret 




15.1 


Rum 


53.6S 


Champagne 




13.8 


Brandy 


53.39 


Gooseberry 




11.84 


Gin . 


51.6 


Elder 




8.79 


Port . 


22.9 


Ale . 




6.87 


Madeira . 


22.27 


Porter 




4.2 


Currant . 


20.55 


Cider 


9.8 to 5.2 


Teneriffe . 


19.79 




Prof 


Brande, 



WEIGHT OF COMPOSITION SHEATHING NAILS. 



Number- 


Length in 


Number in 




Length in 


Number in 




Length in 


Number in 




inches. 


a pound. 




inches. 


a pound. 




inches. 


a pound. 


1 


1 


290 


6 


1 


190 


10 


11 


101 


2 


7 


260 


7 


H 


184 


11 


11 


74 


3 


1 


212 


8 


U 


168 


12 


2 


64 


4 


1| 


201 


9 


H 


110 


13 


2i 


59 


5 1 


u 


199 















CEMENT. 



Ashes 2 T5JiTt^ ^ 

Clav ' 3 " ( Mixed with oil, will resist the weather equal to 

Sand, 1 - S '^^'^^'■ 



HYDRAULIC CEMENT. 

A barrel contains 300 lbs., equal to 4 struck bushels. 



BROWN MORTAR. 



One third Thomaston lime. 

Two thirds sand, and a small quantity of hair. 



MISCELLANEOUS. 



243 



BRICKS, LATHS, ETC. 



Dimensions. 



Common brick 
Front brick . 



8 to 7|x4iX2i inches. 
8i X4^X2^ ** 



20 common bricks to a cubic foot, when laid ; 

15 " " *' a foot of 8 inch wall, when laid. 

Laths are l\ to H inches by four feet in length, are usually set 
i of an inch apart, and a bundle contains 100. 

Stourbridge fire-brick, 9ix4|x2i inches. 



HAY. 



10 cubic yards of meadow hay weigh a ton. When the hay is ta- 
ken out of large or old stacks, 8 and 9 yards will make a ton. 

11 to 12 cubic yards of clover, when dry, weigh a ton. 







HILLS 


IN AN ACRE OF GROUND. 




40 feet 


apart 


27 hills, 


8 feet apart 


680 hills, 


35 '' 






35 




6 " 


1210 *' 


30 " 








48 




5 " 


1742 " 


25 " 








69 




3i " '* . 


3556 " 


20 *' 








108 




3 " " . 


4840 " 


15 " 








193 




2^ " " . 


6969 *' 


12 '' 








302 




2 " *' . 


10890 ^* 


10 " 








435 




1 " " . 


43560 " 



DISPLACEMENT OF ENGLISH VESSELS OF WAR, WHEN LAUNCH- 
ED AND WHEN READY FOR SEA. 



Weight of hull, launched . 
Weight received on board . 

Weight complete . . . . 



120 


80 


74 


Razee. 
50 


52^ 


46 

Tons. 
795 
670 


28 


Corv. 
18 

Tons. 
281 
326 

607 


Tons. 
2467 
2142 

4609 


Tons. 
1882 

1723 


Tons. 
1617 
1359 


Tons. 
1448 
1044 

2492 


Tons. 
1042 
1067 

2109 


Tons. 
413 
370 

783 


3605 


2976 


1465 



Brig. 
IS 

Ton€. 
213 
242 



455 

Edye's JV*. C. 



244* 



MISCELLANEOUS. 



WEIGHT OF LEAD PIPE PER YARD, 
From i to 4-| Inches Diameter. 



Weight ii 


lbs. and oz 


Weight in 


lbs. and oz. 


i inch medium . 3 





H inch extra light 


9 





" strong . 4 





" light 


13 





l' inch light . 3 





" medium . 


15 


8 


" medium . 4 





" strong 


19 


— 


'' strong . 5 





1} inch medium . 


16 


— 


*' extra strong 6 


6 


" strong 


20 


— 


1 inch light . 5 


— 


2 inch ligtit 


16 


12 


" medium . 6 


8 


" medium . 


20 


— 


" strong . 7 


8 


" strong 


23 


— 


" extra strong 8 


4 


2^ inch light 


25 


— 


i inch extra light 5 


— 


" medium . 


30 


— 


" light . 6 


4 


''' strong 


35 


— 


" medium . 8 


— 


3 inch light 


30 


— 


" strong . 9 


12 


" medium . 


35 


— 


" extra strong 10 


8 


" strong 


44 


— 


1 inch extra light 6 


14 


3§ inch medium . 


45 


— 


" hght . 8 


5 


" strong 


54 


— 


*' medium . 10 


5 


" extra strong 


^70 


— 


*' strong . 12 


4 


4 inch waste, light 






li inch extra light 8 


5 


" " medium 


'21 


— 


" light . 9 


12 


'' '' strong, 


26 


— 


" medium . 11 


— 


4i inch '' light, 


— 


— 


'' strong . 12 


8 


" '* medium 


24 


— 


" extra strong 14 


10 


'' " strong, 


29 


— 



Very light Pipe. 



Diametar. 


Weight in 


ibs. and oz. 


Diameter. 


Weight in lbs. and oz. 


i inch 


1 





J inch 


3 6 


§ " 


. H 





1 " 


5 10 


i " 


2 





li - 


6 14 


1 " 


. 2,^ 









TIN. 





Size of sheet. 




Mean thickness. 


Mean weight 
of one sheet. 


Description. 


2^0. oil wire 
gau^^e. 


Thickness of sheet. 


Single . 
Double X 


Inches. 

10x14 
10X14 


31 

27 


Inches. 

.0125, (or 80 to 1 inch) 
.0181, (or 55 to 1 inch) 


Lbs. 

0.5 
0.75 



There are usually 225 sheets in a box. 



MISCELLANEOUS. 



24.& 



RELATIVE PRICES OF AMERICAN WROUGHT IRON. 



Round. 




Square 




4 inches 


27 


4 inches 


27 


3^ to 24 inches . 


26 to 21 


3i to 2i inches 


26 to 20 


2i - i " 


19 


21 - 1 - 


21 




20 to 29 


li" i " 


19 to 26 


i^/a^ 




Hoops. 




i and \ inch to | 


26 to 28 


H to 4 inch 


24 to 33 


i "' 1 " '^ 1X4 


19 






i " i " " iXT%- 


19 to 23 


J5a7Z(i . 


20 



Illustration. — If 4 inch round iron is worth $135 per ton, then 
band iron is worth $100 per ton, for 27 is to 20 as 135 to 100. 



POWER REQUIRED TO PUNCH IRON AND COPPER PLATES-. 



Through an Iron Plate, with a Punch ^ Inch in Diameter, 



.08 inches thick 

.17 " 

.24 " " 



6025 lbs. 
11950 " 
17100 " 



Through a Copper Plate, with a Punch \ Inch in Diameter, 



.08 inches thick 
.17 *' 



3983 lbs. 
7833 " 



The force necessary to punch holes of different diameters through 
metals of various thicknesses, is directly as the diameter of the hole 
and the thickness of the metal. 



To ascertain the Force necessary to Punch Iron or Copper 
Plates, 

Rule.— Multiply, if for iron, 150000, and if for copper, 96000, l^ 
the diameter of the punch and the thickness of the plate, each la 
inches ; the product is the pressure in pounds. 

The use of oil reduces the above 8 per cent. 
X2 



24f6 



WEIGHT OF SQUARE EOLLED IRON. 



IRON. 

Cast Iron expands ywtowo ^^ ^^^ length for one degree of heat ; 
greatest change in the s^hade in this dimate, ytto ^^ i^^ length ; 
exposed to the sun's rays, yoVo 5 shrinks in cooHng from -^^ to ^ 
of its length ; is crushed by a force of 93,000 lbs. upon a square 
inch ; will bear, without permanent alteration, 15,300 lbs. upon a 
square inch, and an extension of y—- ^^ ^^^ length. 

Weight of modulus of elasticity for a base of an inch square, 
18,400,000 lbs. ; height of modulus of elasticity, 5,750,000 feet. 

Wrought Iron expands ytto^ ^^ ^^^' ^^"^^^ ^^^ ^"^ degree of 
heat ; will bear on a square inch, without permanent alteration, 
17,800 lbs., and an extension in length of y^Vo ? cohesive force is 
diminished 30V0 ^Y ^^ increase of 1 degree of heat. 

Weight of modulus of elasticity for a base of an inch square, 
24,920,000 lbs. ; height of modulus of elasticity, 7,550,000 feet. 

Compared with cast iron, its strength is 1.12 times, its extensibility 
0.86 times, and its stiffness 1.3 times. 



WEIGHT OF SQUARE ROLLED IRON, 
From i Inch to 12 Inches, 









AND ONE FOOT IN LENGTH. 






Size in 


Weight in 


Size in 


Weight in 


Size in 


Weight in 


Size in 


Weight in 


iucbbs. 


pounds. 


inches. 


pounds. 


inches. 


pounds. 


inches. 


pounds. 


_!_ 


.013 


2. 


13.520 


4.1 


64.700 


T.i 


190.136 




16^ 


2.1 


15.263 


4.i 


68.448 


1,1 


203.024 




¥ 


.053 


2.i 


17.112 


4.| 


72.305 


8. 


216.336 




1 


.118 


2.t 


19.066 


4.f 


76.264 


8.i 


230.068 




.211 


2.i 


21.120 


4.| 


80.333 


8.i 


244.220 




5 

i 


.475 


2.| 


23.292 


5. 


84.480 


%.% 


258.800 




2.1 


25.560 


5.1 


88 . 784 


9. 


273.792 




i 


.845 


2.1 


27.939 


5.i 


93.168 


9.i 


289.220 




1 


1.320 


3. 


30.416 


5.f 


97.657 


9.i 


305.056 




1 


1.901 


3.8 


33.010 


5.i 


102.240 


9.f 


321.332 




7 


2.588 


3.5 


35.704 


5.1 


106.953 


10. 


337.920 




3.380 


3.f 


38.503 


5.t 


111.756 


10. i 


355.136 




t 


4.278 


3.i 


41.408 


5.1 


116.671 


10.^ 


372.672 




5.280 


3.1 


44.418 


6. 


121.664 


10.^ 


390.628 




3 


6.390 


3.1 


47.534 


6.i 


132.040 


11. 


408.960 




1 


7.604 


3.1 


50.756 


6.^ 


142.816 


11. i 


427.812 




5 


8.926 


4. 


.54.084 


6.i 


154.012 


11. i 


447.024 




'i 


10.352 


4.1 


57.517 


7. 


165.632 


11. i 


466.684 




7 
•8 


11.883 


4.i 


61.055 


7.i 


177.672 


12. 


486.656 



Example.— What is the weight of a bar of rolled iron 1^ inches square and 12 
Inches long 1 * 
In column 1st find H, and opposite to it is 7.604 pounds, which is 7 lbs. and y^^ 



WEIGHT OF ROUND ROLLED IRON. 



247 



of a lb. If the lesser denomination of ounces is required, the result is obtained as 
follows : Multiply the remainder by 16, pointing off the decimals as in multiplica- 
tion of decimals, and the figures remaining on the left of the point indicate the 
number of ounces. 

Thus, -M^ of a lb. = .604 

10 jg 

9.664 ounces. 
The weight, then, is 7 lbs. 9.^^^^ ounces. 

If the weight for less than a foot in length was required, the readiest operation is 
this: 
Example. — What is the weight of a bar 6^: inches square and 9^ inches long ? 
In column 5th, opposite to 6^, is 132.040, which is the weight for a foot in length. 

6^X12 inches = 132.040 

6. " isi = 66.020 
3. " is ^ of 6= 33.010 
•i " is ^ of 3= 5.5016 
4 " is i of i = _2.7508 



WEIGHT OF ROUND ROLLED IRON, 
From 4 Inch to l^ Inches Diameter, 

AND ONE FOOT IN LENGTH. 



Diameter 


Weight in 


Diameter 


Weight in 


Diameter 


Weight in 


Diameter 


Weight in 


in inches. 


pounds. 


in inches. 


pounds. 


in inches. 


pounds. 


in inches. 


pounds. 


3 


.010 


2.i 


11.988 


4.i 


53.760 


7.1 


159.4.56 


.041 


2.1 
2.1 


13.440 
14.975 


4.1 
4.1 


56.788 
59.900 


8. 
84 


169.8.56 
180.696 


•TO 


.119 


2.i 


16.688 


4.| 


63.094 


8.i 


191.808 




.165 


2.| 


18.293 


5. 


66.752 


8.i 


203.260 


•8 


.373 


2.f 


20.076 


5.1 


69.731 


9. 


215.040 


•^ 


.663 


2.| 


21.944 


5.i 


73.172 


9.i 


227.152 


,- 


1.043 


3. 


23.888 


5.f 


76.700 


9.^ 


239.600 


.f 


1.493 


SA 


25.926 


5.i 


80.304 


9.1 


252.376 


.1 


2.032 


3.1 


28.040 


5.1 


84.001 


10. 


266.288 




2.654 


3.f 


30.240 


5.1 


87.776 


10. i 


278.924 


1«8 


3.360 


3.i 


32.512 


5.1 


91.634 


10. i 


292.688 




4.172 


3.1 


34.886 


6. 


95.552 


10. i 


306.800 


1.5 


5.019 


3.1 


37.332 


6.1 


103.704 


11. 


321.216 


l.i 


5.972 


3.1 


39.864 


6.i 


112.160 


11.1 


336.004 




7.010 


4. 


42.464 


6.1 


120.960 


11. i 


351.104 


1 .5 


8.128 


4.1 


45.174 


7. 


130.048 


11. i 


366.536 


1.^ 


9.333 


4.i 


47.952 


7.i 


139.544 


12. 


382.208 


2. 


10.616 


4.f 


50.815 


7.i 


149.328 







The application of this table is precisely similar to that of the preceding one. 



248 



WEIGHT OF FLAT ROLLED IRON. 



WEIGHT OF FLAT ROLLED IRON, 
From ixk Inch /o 5f X6 Inches y 

AND ONE FOOT IN LENGTH. 



Breadth Thic 


fness 


Weight in 


Breadth 


Thickness 


Weight in 


Breadth 


Thickness 


in inchea. iu in 


ches. 


pounds. 


in inches. 


in inches. 


pounds. 


ia inches. 


in inches. 


•i . 


i 


0.211 


I.i 


14 


5.808 


2. 


.1 




0.422 


1.^ 


4 


0.633 




.1 




8 

1 


0.634 




4 


1.266 




•8 


.1 ; 


8 


0.264 






1.900 




• i 




i 


0.528 




^ 


2.535 




• 1 




t 


0.792 




.1 


3.168 




.t 




i 


1.056 




A 


3.802 




7 
•5 


.1 




0.316 




7 
•8 


4.435 








I 


0.633 




1. 


5.069 




1 .1 




.1 


0.950 




1.1 


5.703 




I.i 




■ ■ 


1.265 




I.i 


6.337 




l»t 




'y 


1.584 




l.t 


6.970 




I.i 


•i : 


1 ' 


0.369 


i.| 


.¥ 


0.686 




l.| 






0.738 




•t 


1.372 




l.| 






1.108 






2.059 




l.f 






1.477 




A 


2.746 


2.| 


•8 






1.846 




' 5 


3.432 




•i 






2.217 




I 


4.119 






1. ! 


8 


0.422 




'.i 


4.805 




• i 






0.845 






5.492 




• 1 






1.267 




1 .8 


6.178 




.3 




^ 


1.690 






6.864 




.1 




1 


2.112 




1 ,^ 


7.551 








I 


2.534 




I.i 


8.237 




l'.| 




1 


2.956 


14 


.8 


0.739 






1 i 


■g 


0.475 




,5 


1.479 




l.§ 




z 


0.950 




2 


2.218 




1 .i 






1.425 




A 


2.957 




l.| 




^ 


1.901 






3.696 




I.i 




^ 


2.375 






4.435 


^ 


1 7 
I»8 




I 


2.850 




• 8 


5.178 




2. 




i 


3.326 






5.914 


2.i 




1 




3.802 




1 .8 


6.653 






i.i 


i 


0.528 




1 .4 


7.393 




;f 




f 


1.056 




■, 


8.132 










1.584 




I.i 


8.871 




•1 




^ 


2.112 




1.^ 


9.610 




•i 




1 


2.640 


I.i 




0.792 




• i 




5 


3.168 






1.584 








7 


3.696 




.1 


2.376 




1 .^ 


1 




4.224 




.i 


3.168 




1«^ 


1 




4.7.52 




.1 


3.960 




I.i 


i.i 




0.580 




1 


4.7.52 










1.161 




.? 


5.544 




l.| 




'; 


1.742 






6.336 




l.f 






2.325 




1 1 

1 . 8 


7.129 




1.1 




,. : 


2.904 




I.i 


7.921 




2. 




.i- 


3.484 






8.713 




2.- 




7 

'IS 


4.065 




I.i 


9.505 


3.i 




1 




4.646 




l.| 


10.297 




.? 


1 


'.i 


6.327 




l.| 


11.089 




.1 



pounds. 



WEIGHT OF FLAT ROLLED IRON. 



24.9 



Breadth 


Thickness 


Weight in 


in inches 


in inches 


pounds. 


2.f 


•i 


4.013 




•1 


5.016 




•4 


6.019 




7 
•8 


7.022 




1. 


8.025 




l.f 


9.02S 




l.i 


10.032 




14 


11.035 




l.i 


12.038 




l.| 


13.042 




l.t 


14.045 




1.1 


15.048 




2. 


16.051 




24 


17.054 




2.i 


18.057 


2.i 


.^ 


1.056 




.1 


2.112 




• i 


3.168 




•i 


4.224 




.1 


5.280 




.1 


6.336 




7 
•8 


7.392 




1. 


8.448 




l.i 


9.504 




l.| 


10.560 




i.i 


11.616 




l.i 


12.672 




1.1 


13.728 




1.^ 


14.784 




1.1 


15.840 




2. 


16.896 




2.1 


17.952 




2.i 


19.008 




2.^ 


20.064 


2-1 


.¥ 


1.109 




.? 


2.218 




.- 


3.327 




.i 


4.436 




i 


5.545 




• ? 


6.654 




• 8 


7.763 




1. 


8.872 




1.1 


9.981 




l.i 


11.090 




1.^ 


12 199 




l.i 


13.308 




l.| 


14.417 




l.i 


15.526 




1-J 


16.635 




2. 


17.744 




2.1 


18.853 




2.i 


19.962 




2.| 


21.071 




2.i 


22.180 




4 


1.162 



Table — (Continued). 

Breadth Thickness Weight in 
in inches, in inches pounds. 



2.2 



•i 

3 
•8 

•i 



•4 
7 
.? 
1. 

1.1 
l.i 

l.i 

l.i 

1.1 

l.£ 

l.| 

2. 

2.i 

2.i 

2.^ 

2.i 

2.1 



1. 

l.i 
l.i 
l.f 
l.i 
1.1 

lA 
1.1 

2. 

2.i 

2.i 

2.i 

2.i 

2.1 

2.^ 



1. 
l.i 
l.i 
l.f 

1.1 



2.323 

3.485 

4.647 

5.808 

6.970 

8.132 

9.294 

10.455 

11.617 

12.779 

13.940 

15.102 

16.264 

17.425 

18.587 

19.749 

20.910 

22.072 

23.234 

24.395 

1.215 

2.429 

3.644 

4.858 

6.072 

7.287 

8.502 

9.716 

10.931 

12.145 

13.360 

14.574 

15.789 

17.003 

18.218 

19.432 

20.647 

21.861 

23.076 

24.290 

25.505 

26.719 

1.267 

2.535 

3.802 

5.069 

6.337 

7.604 

8.871 

10.138 

11.406 

12.673 

13.940 

15.208 

1^.475 



Breadth 


Thickness 


Weight ia 


in inches. 


in inches 


pounds. 


3. 


l.£ 


17.742 




1.1 


19.010 




2. 


20.277 




2.i 


22.811 




2.i 


25.346 




2.;: 


27.881 


3.i 


• V 


1.373 




.:: 


2.746 






4.119 




.1- 


5.492 




• 1 


6.865 




.4 


8.237 




.1 


9.610 




1. 


10.983 




l.i 


12.356 




l.i 


13.730 




l.i 


15.102 




l.i 


16.475 




l.f 


17.848 




1.5 


19.221 




l.| 


20.594 




2. 


21.967 




2.i 


24.712 




2.i 


27.458 




2.^ 


30.204 




3.^ 


32.950 


3.i 


• i? 


1.479 




.:: 


2.957 




I 


4.436 






5.914 




• z 


7.393 




8.871 




.8 


10.350 




1. 


11.828 




l.i 


13.307 




l.i 


14.785 




l.f 


16.264 




l.i 


17.742 




1.1 


19.221 




l.i 


20.699 




l.f 


22.178 




2. 


23.656 




2.i 


26.613 




2.i 


29.570 




2.1 


32.527 




3. 


35.485 




3.i 


38.441 


3.i 


.i 


1.584 




.i 


3.168 


*" 


*! 


4.752 






6.. 336 




•1 


7.921 




A 


9.505 




7 
.8 


11.089 




I. 


12.673 



250 



WEIGHT OF FLAT ROLLED IRON. 



Table — (Continued). 



Breadth 


rhickness 


Weight in 


Breadth 


Thickness 


Weight in 


Breadth 


Thickness 


Weight in 


in inches. 


n inches. 


pounds. 


n inches. 


n inches. 


pounds. 


n inches. 


in inches 


pounds. 


3. J 


l.| 


14.257 


4.i 


2.i 


34.217 


5.i 


2.i 


44.355 


1.- 


15.841 




2.i 


38.019 




2.5 


48.791 




17.425 




2.^ 


41.820 




3. 


53.226 




1,- 


19.009 




3. 


45.623 




3.i 


57 . 662 




20.594 




3.i 


49.425 




3.i 


62.097 




1 ^ 


22.178 




3.i 


53.226 




3.5 


66.533 




23.762 




s.i 


57.028 




4. 


70.968 




J. • g 
2. 


25.346 




4. 


60.830 




4.i 


75.404 




2.i 
2.*- 


28.514 




4.^ 


64.632 




^•t 


79.839 




31.682 


44 


.i 


4.013 




4.5 


84.275 
88.710 




2.1 
3. 


34.851 




.i 


8.026 




5. 




38.019 




.1 


12.039 


5.^ 


i 


4.647 




1:1 


41.187 
44.355 




1. 
l.i 


16.052 
20.066 






9.294 
13.940 


4. 


.2 


1.690 
3.380 




l.i 
l.f 


24.079 

28.092 




l.i 


18.587 
23.234 




6.759 




2. 


32.105 




l.i 


27.881 




10.138 




2.i 


36.118 




1.5 


32.527 




• 4 
1. 


13.518 




2.i 


40.131 




2. 


37.174 




1.1 


16.897 




2.i 


44.144 




2.i 


41.821 




l.i 
14 


20.277 




3. 


48.157 




2.i 


46.468 




23.656 




3.1 


52.170 




2.i 


51.114 




2. 


27.036 




3.i 


56.184 




3. 


55.761 




2.i 


30.415 




3.1 


60.197 




3.i 


60.408 




2.i 


33.795 




4. 


64.210 




3.i 


65.055 




2.1 


37.174 




4.i 


68.223 




3.1 


69.701 




<4( • 4- 

3. 


40.554 




4.i 


72.235 




4. 


74.348 




3.i 

3.1 


43.933 


5. 


.z 


4.224 




4.i 


78.995 




47.313 




.i 


8.449 




4.i 


83.642 




3.5 


50.692 




.f 


12.673 




4.| 


88.288 


44 


1 


1.795 




1. 


16.897 




5. 


92.935 


. 8 


3.591 




l.i 


21.122 




5.i 


97.582 






7.181 




l.i 


25.346 


5.i 


•i 


4.858 




'3 


10.772 




l.i 


29.570 




.i 


9.716 




r. 


14.364 




2. 


33.795 




.i 


14.574 




l.i 


17.953 




2.i 


38.019 




1. 


19.432 




l.i 


21.544 




2.i 


42.243 




l.i 


24.290 




1.5 


25.135 




2.1 


46.468 




r.i 


29.148 




2. 


28 . 725 




3. 


50.692 




1.5 


34.006 




2.i 


32.316 




3.i 


54.916 




2. 


38.864 




2.i 


35.907 




3.i 


59.140 




2.i 


43.722 




2.5 


39.497 




3..I 


63.365 




2.i 


48.580 




3. 


43.088 




4. 


67.589 




2.1 


53.437 




3.5 


46.679 




4.i 


71.813 




3. 


58.296 




3.1 

4.. 


50.269 




4.i 


76.038 




3.i 


63.154 




53.860 




4.5 


80.262 




3.i 


68.012 




57.450 


5.i 


• i 


4.436 




3.1 


72.870 


4.i 


^.i 


3.802 




.i 


8.871 




4. 


77.728 




7.604 




.5 


13.307 




4.i 


82.585 




11.406 




1. 


17.742 




4.i 


87.443 




1 . 


15.208 




l.i 


22.178 




4.1 


92.301 




1.^ 


19.010 




l.i 


26.613 




5. 


97.159 




1.? 


22.812 




14 


31.049 




5.i 


102.017 




26.614 




2. 


35.484 




5.i 


106.876 




2. . 


30.415 




2.i 


39.920 




6. 


116.592 



WEIGHT OF FLAT ROLLED IRON. 251 

Examples.— What is the weight of a bar of iron 5^ inches in breadth by ^ inches 
thick 1 

In column 4, page 250, find 54:, and below it, in column 5, ^ ; and opposite to that 
is 13.307, which is 13 lbs. and -^^ of a lb. 

For parts of a lb. and of a foot, operate precisely similar to the rule laid down for 
table, page 247. 



WEIGHTS OF A SQUARE FOOT OF IRON IN AVOIRDUPOIS POUNDS. 

THICKNESS BY WIRE GAUGE. 

No. on gauge . 1 |2|3|4|5|6|7|8|9|10|11 
Pounds . . 12.5 I 12 I 11 1 10 1 9 I 8 I 7.5 I 7 I 6 I 5.68 1 5 

No. on gauge .12 I 13 I 14 I 15 I 10 i 17 I 18 | 19 1 20 l 21 I 22 
Pounds . . 4.62 1 4.31 I 4 I 3.95 I 3 I 2.5 I 2.18 I 1.93 I 1.62 I 1.5 I 1.37 

Number 1 is y^^, number 4 is J, and number 11 is ^ of an inch. 



CAST IRON. 

To ascertain the weight of a cast iron Bar or Rod, find the weight of a UTOUght 
iron bar or rod of the same dimensions in the preceding tables, and from the weight 
deduct the ^^i^ th part ; or say. 

As 486.65 : 450.55 : : the weight in the table : to the weight required. Thu^ 

What is the weight of a piece of cast iron 4X3|X12 inches 1 
In table, page 250, the weight of a piece of wrought iron of these dimensions is 
50.692 lbs. 

486.65 : 450.55 : : 50.692 : 46.93 lbs. 

Or, by an easier mode, though not so minutely correct, 
As 281 : 260 : : 50.692 : 46.90 lbs. 



To find the Weight of a piece of Cast or Wrought Iron of any 
size or shape. 

By the rules given in Mensuration of Solids (see page 81), ascertain the number 
of cubic inches in the piece, multiply by the weight of a cubic inch, and the 
product will be the weight in pounds. 

EXAMPLES. 

What is the weight of a block of wrought iron 10 inches square by 15 inches io 
length 1 

10X10X15 = 1500 cubic inches. 

.2816 weight of a cubic inch. 
422.4000 pounds. 

What is the weight of a cast iron bail 15 inches in diameter ? 
By table, page 255= 176.7149 cubic inches. 

.2607 Weight of a cubic inch. 
460.6957 pounds. 



252 



WEIGHT OF CAST IRON PIPES. 



WEIGHT OF CAST IRON PIPES OF DIFFERENT THICKNESSES, 
From 1 Inch to 36 Inches Bore, 



AND ONE FOOT IN LENGTH. 



Bore. Thi 


,kness 


; Weight. 


Bore. ITbic 


kness 


Weight. 


Bore. Thi 


:kness 


Weight. 


Inches. Inc 


hes. 


Pounds. 


Inches. Inc 


hes. 


Pounds. 


Inches. In 


:hes. 


Pounds. 


1. 


;| 


3.06 


6. 


4 


49.60 


11. i 


4 


58.82 




5.05 




, ; 


58.96 




4 


74.28 


14 


4 


3.67 


6.i 


Y 


34.32 




4 


90.06 




4 


6. 




- 


43.68 




4 


106.14 


1.^ 


4 


6.89 






53.30 


1 




122.62 




4 


9.80 




- 


63.18 


12. 


'i 


61.26 


14 


2 


7.80 


7. 


1 


36.66 




1 


77.36 




* 


11.04 


1 


1 


46.80 




f 


93.70 


2. 


1 


8.74 


i 


5 


56.96 




i 


110.48 




i 


12.23 


I 


1 


67.60 


1 




127.42 


24 


3 


9.65 


1 




78.39 


12. i 


i 


63.70 




i 


13.48 


74 


i 


39.22 




1 


80.40 


24 


1 


10.57 




1 


49.92 




i 


97.40 






14.66 




1 


60.48 




7 
8 


114.72 




1 


19.05 




1 


71.76 


1 




132.35 


24 


1 


11.54 


l' 




83.28 


13. 


i 


66.14 




1 


15.91 


8. i 


i 


41.64 




1 


83.46 




1 


20.59 




5 


52.68 




i 


101.08 


3. 


f 


12.28 




1 


64.27 




7 
8 


118.97 




i 


17.15 




■g 


76.12 


1 




137.28 




1 


22.15 


1, 




88.20 


13. f 


i 


68.64 




i 


27.56 


8.i 


'k 


44.11 




1 


86.55 


34 


1 


18.40 




5 
8 


56.16 




^ 


104.76 






23.72 




i 


68. 




7 
8 


123.30 




1 


20.64 




7 
8 


80.50 


1 




142.16 


34 


1 


19.66 


1 




93.28 


14. 


i 


71.07 






25.27 


9. 


1 


46.50 




1 


89.61 




1 


31.20 






58.92 




i 


108.46 


34 


1 


20.90 




£ 


71.70 




1 


127.60 






26.83 




7 


84.70 


1 




147.03 




2 


33.07 


1 




97.98 


14.^ . 


i 


73.72 


4. 


i 


22.05 


9.^ 


i 


48.98 


. 


f 


92.66 




1 


28.28 




1 


62.02 


j 


i 


112.10 




i 


34.94 


i 


i 


75.32 


1 


I 


131.86 


44 


i 


23.35 


! 


1 


88.98 


i 1 




151.92 






29.85 


1 




102.90 


15. 1 . 


i 


75.96 




2 


36.73 


10. 


^ 


51.46 


1 


1 


95.72 


44 


1 


24.49 


1 


1 


65.08 




f 


115.78 




1 


31.40 


1 


1 


78.99 




i 


136.15 




I 


38.58 


1 


i 


93.24 


, 1 1 




156.82 


44 


1 


25.70 


1 




108.84 


15. i i 


1 


78.40 






32.91 


104 


1 


53.88 






98.78 




1 


40.43 






68.14 




% 


119.48 


5. 


^ 


26.94 




4 


82.68 




1 


140.40 




1 


34.34 




4 


97.44 


1 




161.82 




1 


42.28 


1 




112.68 


16. 


1 


80.87 


54 




29.40 


11. 


4 


56.34 






101.82 




1 


37.44 






71.19 




I 


123.14 




4 


45.94 




4 


86.40 




1 


144.76 


6. 


1 


31.82 


1 


• t 


101.83 


1 




166.60 




4 


40.56 


; 1 




117.60 


16.^ 


■i 


83.30 



WEIGHT OF CAST IRON PIPES. 



253 









TABLE~(Continued ). 








Bore. Thic 


inesi 


Weight. 


Bore. Thic 


kness 


Weight. 


Bore. jThic 


kness 


Weight. 


Inches. Inc 


les. 


Pouads. 


Inches. Inc 


hes. 


Pounds. 


Inches. Inc 


hes. 


Pounds. 


16. i 




104.82 


22. 


5 
8 


138.60 


30. 1. 




303.86 




7 


126.79 




5 


167.24 




1 


343.20 




149.02 




7 
3 


196.46 


31. '. 


f 


233.40 


i! 


8 


171.60 






225.38 




i 


273.40 


17. 


1 


85.73 


23. 


5 


144.77 






313.68 




107.96 




2 


174.62 




i 


354.24 




I 


130.48 




_ 


204.78 


32. 


1 


240.76 




1 


153.30 






235.28 




1 


281.94 


1 


^ 


176.58 


24. 


1 


150.85 






323.49 


17. i 


1 


88.23 
111.06 




g 


181.92 

213.28 


33. 


1 


365.29 
248.10 




I 


134.16 






245.08 




7 


290.50 




4- 
7 


157.59 


25. 


1 


156.97 






333.24 


1 


8 


181.33 




i 


189.28 




i 


376.26 


18. 


1 


114.10 




1 


221.94 


34. 


i 


255.45 




8 


137.84 






254.86 




i 


298.88 




1 


161.90 


26. 


i 


196.62 






342.88 


1 


8 


186.24 




1 


230.56 




I 


387.13 


19. 




120.24 






264.66 




i 


431.76 




•■ 


145.20 


27. 


I 


204.04 


35. 


I 


262.70 




170.47 




7 


239.08 




7 
3 


307.62 


1 


8 


195.92 






274.56 






352.86 


20. 


5 


126.33 


28. 


1 


211.32 




i 


39S.10 




7 


1 52 . 53 




8 


247.62 




? 


443.96 




179.02 






284.28 


36. 


i 


270.18 


1 


• 8 


205.80 


29. 


t 


218.70 




i 


316.36 


21. 


4 


132.50 




7 
8 


256.20 






362.86 






159.84 






294.02 




'•I 


409.34 




• ? 
7 

• 8 


187.60 


30. 


'i 


226.20 




• i 


456.46 


1 




215.52 




.1 


264.79 









Note.— These weights do not include any allowance for spigot and faucet ends. 

Y 



254! WEIGHT OF A SQUARE FOOT OF CAST IRON, ETC. 



WEIGHT OF A SQUARE FOOT OF CAST AND WROUGHT IRON, 
COPPER, AND LEAD, 

From ^th to 2 Inches thick. 



Thickness. 


Cast Iron. 


Wrought Iron. 
Hard rolled. 


Copper. 
Hard roiled. 


Lead. 




Pounds. 


Pounds. 


Pounds. 


Pounds. 


•tV 


2.346 


2.517 


2.890 


3.691 


■i 


4.693 


5.035 


6.781 


7.382 


.1% 


7.039 


7.552 


8.672 


11.074 


• i 


9.386 


10.070 


11.562 


14.765 


•A 


11.733 


12.588 


14.453 


18.456 


.§ 


14.079 


15.106 


17.344 


22.148 


-T^ 


16.426 


17.623 


20.234 


25.839 


• i 


18.773 


20.141 


23.125 


29.530 


9 

•TS 


21.119 


22.659 


26.016 


33.222 


•1 


23.466 


25.176 


28.906 


36.913 


11 

•Tff 


25.812 


27.694 


31.797 


40.604 


4 


28.159 


30.211 


34.688 


44.296 


•H 


30.505 


32.729 


37.578 


47.987 


7 

"5 


32.852 


35.247 


40.469 


51.678 


•\i 


35.199 


37.764 


43.359 


55.370 


1 inch 


37.545 


40.282 


46.250 


59.061 


1.1 


42.238 


45.317 


52.031 


66.444 


l.i 


46.931 


50.352 


57.813 


73.826 


i.i 


51.625 


55.387 


63.594 


81.210 


14 


56.317 


60.422 


i 
69.375 


88.592 


1.1 


61.011 


65.458 


75.156 


95.975 


i.i 


65.704 


70.493 


80.938 


103.358 


i.| 


70.397 


75.528 


86.719 


110.740 


2. 


75.090 


80.563 


92.500 


118.128 



Note.— The Specific Gravity of the Wrought Iron is that of Pennsylvania 
plates, and of the Copper, that of plates from the works of Messrs. Phelps, Dodge 
& Co., in Connecticut. The Lead, a mean from several places. 



WEIGHT AND CAPACITY OF CAST IKON AND LEAD BALLS. 25& 



WEIGHT AND CAPACITY OF CAST IRON AND 
LEAD BALLS, 

From 1 to 20 Inches in Diameter. 



Diameter in 
inches. 


Capacity in cubic 
inches. 


CAST IRON. 
Pounds. 


LEAD. 

Pounds. 


1. 


.5235 


.1365 


.2147 


l.i 


1.7671 


.4607 


.7248 


2. 


4.1887 


1.0920 


1.7180 


2.^ 


8.1812 


2.1328 


3.3554 


3. 


14.1371 


3.6855 


5.7982 


3.i 


22.4492 


5.8525 


9.2073 


4. 


33.5103 


8.7361 


13.744 


4.i 


47.7129 


12.4387 


19.569 


5. 


65.4498 


17.0628 


26.843 


5.i 


87.1137 


22.7206 


35.729 


6. 


113.0973 


29.4845 


46.385 


6.i 


143.7932 


37.4528 


58.976 


7. 


179.5943 


46.8203 


73.659 


7.^ 


220.8932 


57.5870 


90.598 


8. 


268.0825 


69.8892 


109.952 


8.i 


321.5550 


83.8396 


131.883 


9. 


381.7034 


99.5103 


156.553 


9.i 


448.9204 


117.0338 


184.121 


10. 


523.5987 


136.5025 


214.749 


11. 


696.9098 


181.7648 


285,832 


^ 12. 


904.7784 


235.8763 


371.096 


13. 


1150.346 


299.6230 


471.806 


14. 


1436.754 


374.5629 


589.273 


15. 


1767.145 


460.6959 


724.781 


16, 


2144.660 


559.1142 


879.616 


17. 


2572.440 


670.7168 


10.55.066 


18. 


3053.627 


796.0825 


1252.422 


19. 


3591.363 


936.2708 


1472.970 


20. 


4188.790 


1U92.0200 


1717.995 



256 



WEIGHT OF COPPER RODS AND PIPES. 



WEIGHT OF COPPER EODS OR BOLTS, 

From i to 4: Inches in Diameter, 



•TF 

3 
• ¥ 

•1 6 



9 
•T6- 

5. 

• 8 
1 1 

•rg- 

3 

• 4 
13 

•TF 

7 

• ¥ 



AND ONE FOOT IN LENGTH. 



Pounds, 



.1892 

.2956 

.4256 

.5794 

.7567 

.9578 

1.1824 

1.4307 

1.7027 

1.9982 

2.3176 

2.6605 

3.0270 





1 
YE 


3.4170 




1 
8 


3.8312 




3 

TF 


4.2688 




1 

4 


4.7298 




TF 


5.2140 




3 

8 


5.7228 




7 
TF 


6.2547 




1 
2 


6.8109 




T% 


7.3898 




3 


7.9931 




3 
4 


9.2702 




7 
8- 


10.6420 


2. 




12.1082 



2.1 


13.6677 


2.i 


15.3251 


2.t 


17.0750 


2.i 


18.9161 


2.1 


20.8562 


2.1 


22.8913 


2.1 


25.0188 


3. 


27.2435 


3.1 


29.5594 


34 


31.9722 


3.f 


34.4815 


3.^ 


37.0808 


3.1 


39 . 7774 


3.i 


42.5680 


3.1 


45.4550 


4. 


48.4330 



WEIGHT OF RIVETED COPPER PIPES, 

From 5 to 30 Inches in Diameter, from 3 to j^ths thick, 



AND ONE FOOT IN LENGTH. 



Diam. 


Thickness 


Weight in 


Diam. 


Thickness 


Weight in 


Diam. 


Thickness 


Weight in 


in ins. 


in 16;hs, 


pounds. 


in ins. 


in 16ihs. 


pounds. 

30.598 


in ins. 


in I6ths. 


pounds. 


5. 


3 


12.497 


9.i 


4 


19. 


4 


60.142 


5. 


4 


16.880 


10. 


4 


32.208 


19. 


5 


75.233 


5.i 


3 


13.628 


11. 


4 


35.200 


20. 


5 


78.208 


5.i 


4 


18.395 


12. 


4 


38.456 


21. 


5 


82.984 


6. 


3 


14.765 


13. 


4 


41.456 


22. 


5 


86.771 


6. 


4 


19.908 


14. 


4 


44.640 


23. 


5 


90.571 


6.^ 


3 


15.897 


15. 


4 


47.646 


24. 


5 


94.308 


6.i 


4 


21.415 


15. 


5 


59.588 


25. 


5 


98.122 


7. 


3 


17.034 


16. 


4 


.50.752 


26. 


5 


101.897 


7. 


4 


22.932 


16. 


5 


63.470 


27. 


5 


105.700 


7.-^ 


4 


24.447 


17. 


4 


53.856 


28. 


5 


109.446 


8. 


4 


25.961 


17. 


5 


67.. 344 


29. 


5 


113.221 


8.-^. 


4 


27.471 


18. 


4 


57.037 


30. 


5 


116.997 


9. 


4 


28.985 


18. 


5 


71.258 









The above weights include the laps on the sheets for riveting and caulking. 

The weights of the rivets are not 
depends upon the distance they are 
ter of the pipe. 



added; the number per lineal foot of pipe 
placed apart, and their size upon the diame- 



COPPER, LEAD, AND BRASS. 257 



COPPER. 

To ascertain the Weight of Copper. 

Rule. — Find by calculation the number of cubic inches in the piece, multiply 
them by .32118, and the product will be the weight in pounds. 

Example. — What is the weight of a copper plate ^ an inch thick by 16 inches 
square 1 

162 = 256 

.5 for ^ an inch. 
128.0X. 32118 = 41.111 pounds. 



LEAD. 

To ascertain the Weight of Lead, 

Rule. — ^Find by calculation the nimiber of cubic inches in the piece, and multi' 
ply the sum by .41015, and the product will be the weight in pounds. 

Example.— What is the weight of a leaden pipe 12 feet long, 3| inches in diam- 
eter, and 1 inch thick ? 

By rule in Mensuration of Surfaces^ to ascertain the area of cylindrical rings f 

Area of (3H-1+1) = 25.967 
" " si = 11.044 

Difference, 14.923, or area of ring. 

144 = 12 feet. 

2148.912X.41015 = 881.376 pounds. 



BRASS. 

To ascertain the Weight of ordinary Brass Castings, 

Rule.— Find the number of cubic inches in the piece, multiply by .3112, and the 
product will be the weight in pounds. 

Y2 



258 



CABLES AND ANCHORS. 



CABLES AND ANCHORS. 



Table 


showing 


le Size of 


Cables and Anchors proportioned to the 






Tonnage of Vessels. 






Tonnage of 
vessel. 


Cables. 
Circumference 


Chain Cables. 
Diameter in 


Proof in 


Weisht of 
Anchor in 


Weight of a 
fathom of 


Weight of a 
fathom of 




in inches. 


inches. 




pounds. 


Chain. 


Cable. 


6 


3. 


•1% 


•I 


56 


5.i 


2.1 


8 


4. 


3 
• 8 


14 


84 


8. 


4. 


10 


4.i 


■^ 


2.i 


112 


11. 


4.6 


15 


5.1- 


• i 


4. 


168 


14. 


6.5 


25 


6. 


•1 


5. 


224 


17. 


8.4 


40 


6.^ 


.f 


6. 


336 


24. 


9.8 


60 


7. 


•rk 


7. 


392 


27. 


11.4 


75 


7.i 


3 

• 4 


9. 


532 


30. 


13. 


100 


8. 


■H 


10. 


616 


36. 


15. 


130 


9. 


7 


12. 


700 


42. 


18.9 


150 


9.^ 


■U 


14. 


840 


50. 


21. 


180 


10. i 


1. 


16. 


953 


56. 


25.7 


200 


11. 


i-tV 


18. 


1176 


60. 


28.2 


240 


12. 


i-i 


20. 


1400 


70. 


33.6 


270 


12. i 


l-T^ 


21. 


1456 


78. 


36.4 


320 


13.^ 


l-i 


22. i 


1680 


86. 


42.5 


360 


14. 


1-A 


25. 


1904 


96. 


45.7 


400 


14. i 


1-1 


27. 


2072 


104. 


49. 


440 


15.^ 


1-tV 


30. 


2240 


115. 


56. 


480 


16. 


l.i 


33. 


2408 


125. 


59.5 


520 


16.^ 


1-T^ 


36. 


2800 


136. 


63.4 


570 


17. 


1-1 


39. 


3360 


144. 


67.2 


620 


17. i 


1-U 


42. 


3920 


152. 


71.1 


680 


18. 


1-f 


45. 


4200 


161. 


75.6 


740 


19. 


l-H 


49. 


4480 


172. 


84.2 


820 


20. 


I-i 


52. 


5600 


184. 


93.3 


900 


22. 


1-il 


56. 


6720 


196. 


112.9 


1000 


24. 


2. 


60. 


7168 


208. 


134.6 



The proof in the U. S. Naval Service is about 12^ per cent, less than the above. 

The utmost strength of a good hemp rope is 6400 lbs. to the square inch ; in prac- 
tice it should not be subjected to more than half this strain. It stretches from 4 to 
•^, and its diameter is diminished from 5^ to ^ before breaking. 

A difference in the quality of hemp may produce a difference of ^ in the strength 
of ropes of the same size. 
The strength of Manilla is about J that of hemp. 
White ropes are one third more durable. 



CABLES. 



259 



CABLES. 

Table showing what Weight a good Hemp Cable will bear with 

Safety. 



' Circumference. 


Pounds. 


Circumference. 


Pounds. 


Circumference. 


Pounds. 


6. 


4320. 


10.25 


12607.5 


14.50 


25230. 


6.25 


4687.5 


10.50 


13230. 


14.75 


26107.5 


6.50 


5070. 


10.75 


13867.5 


15. 


27000. 


6.75 


5467.5 


11. 


14520. 


15.25 


27907.5 


7. 


5880. 


11.25 


15187.5 


15.50 


28830. 


7.25 


6307.5 


11.50 


15870. 


15.75 


29767.5 


7.50 


6750. 


11.75 


16567.5 


16. 


30720. 


7.75 


7207.5 


12. 


17280. 


16.25 


31687.5 


8. 


7680. 


12.25 


18007.5 


16.50 


32670. 


8.25 


8167.5 


12.50 


18750. 


16.75 


33667.5 


8.50 


8670. 


12.75 


19507.5 


17. 


34680. 


8.75 


9187.5 


13. 


20280. 


17.25 


35707.5 


9. 


9720. 


13.25 


21067.5 


17.50 


36750. 


9.25 


10267.5 


13.50 


21870. 


17.75 


37807.5 


9.50 


10830. 


13.75 


22687.5 


18. 


38880. 


9.75 


11407.5 


14. 


23520. 


18.25 


39967.5 


10. 


12000. 


14.25 


24367.5 







To ascertain the Strength of Cables. 

Multiply the square of the circumference in inches by 120, and the product is 
the weight the cable will bear in pounds, with safety. 



To ascertain the Weight of Manilla Ropes and Hawsers. 

Multiply the square of the circumference in inches by .03, and the product is 
the weight in pounds of a foot in length. 
This is but an approximation, and yet it is sufficiently correct for many purposes. 



Table showing what Weight a Hemp Rope will bear with Safety. 



Circumference. 


Pounds, 


Circumference. 


Founds. 


Circumference. 


Poundi. 


1. 


200. 


3.i 


2450. 


6. 


7200. 


i.i 


312.5 


3.1 


2812.5 


6.i 


7812.5 


1.4 


450. 


4. 


3200. 


6.1 


8450. 


l.J 


612.5 


44 


3612.5 


6.1 


9112.5 


2. 


800. 


4.i 


4050. 


7. 


9800. 


2.i 


1012.5 


• 4.i 


4512.5 


7.{ 


10512.5 


2.i 


1250. 


5. 


5000. 


7.i 


11250. 


2.1 


1512.5 


5.1 


5512.5 


7.1 


12012.5 


3. 


1800. 


5.i 


6050. 


8. 


12800. 


3.i 


2112.5 


5.1 


6612.5 







260 CABLES. 



To ascertain the Strength of Ropes, 

Multiply the square of the circumference in inches by 200, and it gives the 
weight the rope \n\\ bear in pounds, with safety. 

To ascertain the Weight of Cable-laid Ropes. 

Multiply the square of the circumference in inches by .036, and the product is 
the weight in pounds of a foot in lengtli. 



To ascertain the Weight of Tarred Ropes and Cables. 

Multiply the square of the circumference by 2.13, and divide by 9 ; the product 
is the weight of a fathom in pounds. 

Or, multiply the square of the circumference by .04, and the product is the 
weight of a foot. 

For the ultimate strength, divide the square of the circumference in inches by 
5 ; the product is the weight in tons. 

A square inch of hemp fibres will support a weight of 92000 lbs. 



BLOWING ENGINES. 261 



BLOWING ENGINES. 

The object of a blast is to supply oxygen to furnaces. 

The quantity of oxygen in the same bulk of air is different at dif- 
ferent temperatures. Thus, dry air at 85° contains 10 per cent, less 
oxygen than when at the temperature of 32^ ; when saturated with 
vapour, it contains 12 per cent. less. 

Hence, if an average supply of 1500 cubic feet per minute is re- 
quired in winter, 1650 feet will be required in summer. 

The pressure ordinarily required for smelting purposes is equal to 

a column of mercury from 3 to 7 inches. 

The capacity of the Reservoir should exceed that of the cylinder or cylinders, and 
the area of the pipes leading to it should be ^-^ of the area of the cylinder. 

The quantity of air at atmospheric density delivered into the reservoir, in conse- 
quence of escapes through the valves, and the partial vacuum necessary to produce 
a current, will be about j less than the capacity of the cylinder. 

To find the Poiver ivhen the Cylinder is Double Acting, 
Let P represent pressure in lbs. per square inch, 

V " the velocity of the piston in feet per minute, 
a " the area of the cylinder in inches, 
1.25 " the friction necessary to work the machinery. 
Then Fva 1.25 = the power in lbs. raised 1 foot high per minute. 

^ , „^ When Single Acting. 

Pi?al.25 ^ ^ 

— = the power in lbs. raised 1 foot high per minute. 

Air expands nearly 2| times its bulk while in the fire of an ordinary furnace. 

Dimensions of a Fui^nace, Engines, SfC, 

Furnace. At Lonakoning (Md.). Diameter at the boshes 14 feet, which fall 
in, 6.33 inches in every foot rise. 

Engine. Diameter of cylinder 18 inches, length of stroke 8 feet. 

Averaging 12 revolutions per minute, with a pressure of 50 lbs. per square inch. 

Boilers, Five : each 24 feet in length, and 36 inches in diameter. 

Blast Cylinders. 5 feet diameter, and 8 feet stroke. 

At a pressure of from 2 to 2^ lbs. per square inch, the quantity of blast is 3770 
cubic feet per minute, requiring a power of about 50 horses to supply it. 

180 tons air is required to make 10 tons pig iron, and burn the coke from 50 tons 
coal. The ore yielding about 33 per cent, of iron. 

Steam Boilers. Two cylinders, 12 inches in diameter and 12 inches stroke, aided 
by exhausting into a condenser, and with steam of 30 lbs. pressure per square inch, 
will make 50 revolutions per minute, and drive 4 blowers, each 54 inches in diame- 
ter, and 30 inches wide, 300 revolutions in a minute, furnishing the necessary blast 
for burning anthracite coal on a grate surface of 108 square feet ; supplying 4400 
cubic feet steam per minute, at a pressure of 30 lbs. per square inch. 

35 cubic feet of steam used in the cylinder of a blowing engine will drive 
blowers 4 feet in diameter by 26 inches face, and furnish the necessary blast to 
an anthracite fire, for generating 1150 cubic feet steam, the time 1 minute, and 
pressure per square inch 35 lbs. 



262 MISCELLANEOUS NOTES. 

MISCELLANEOUS NOTES. 



ON MATERIALS, ETC. 



Wood is from 7 to 20 times stronger transversely than longitu- 
dinally. 

In Buffon's experiments, b, d, and I being the breadth, depth, and 
length of a piece of oak in inches, the weight that broke it in pounds 

wa3M^(51pio). 

The hardness of metals is as follows : Iron, Platina, Copper, Silver 
Gold, Tin, Lead. ■ tt , 

A piece spliced on to strengthen a beam should be on its convex 
side. 

Springs are weakened by use, but recover their strength if laid by. 

A pipe of cast iron 15 inches in diameter and .75 inches thick v/ill 
sustain a head of water of 600 feet. One of oa^, 2 inches thick, and 
of the same diameter, will sustain a head of 180 feet. 

When the cohesion is the same, the thickness varies as the height 
X the diameter. 

When one beam is let in, at an inclination to the depth of another, 
so as to bear in the direction of the fibres of the beam that is cut ; 
the depth of the cut at right angles to the fibres should not be more 
than i of the length of the piece, the fibres of which, by their cohe- 
sion, resist the pressure. 

Metals have five degrees of lustre— splendent, shining, glistening, 
glimmering, and dull. 

The Vernier Scale is |^, divided into 10 equal parts ; so that it 
divides a scale of lOths into lOOths w^hen the lines meet even in 
the two scales. 

A luminous point, to produce a visual circle, must have a velocity of 10 feet in a 
second, the diameter not exceeding 15 inches. 

Tides. The difference in time between high water averages 
about 49 minutes each day. 

In Sandy soil, the greatest force of a pile-driver will not drive a pile over 15 feet. 

A fall of yV of an inch in a mile will produce a current in rivers. 

Melted snow produces about | of its bulk of w^ater. 

All solid bodies become luminous at 800 degrees of heat. 

At the depth of 45 feet, the temperature of the earth is uniform throughout the 
year. 

A Spermaceti candle .85 of an inch in diameter consumes an inch Ik lent^th in 1 
hour. " 

Silica vi the base of the mineral world, and Carbon of the organized. 

Sound passes m water at a velocity of 4708 feet per second. 



MISCELLANEOUS NOTES. 263 

SOLDERS. 

For Lead, melt 1 part of Block tin ; and when in a state of fusion, 
add 2 parts of Lead. Resin should be used with this solder. 

For Tin, Pewter 4 parts, Tin ], and Bismuth 1 ; melt them to- 
gether. Resin is also used with this solder. 

For Iron, tough Brass, with a $mall quantity of Borax. 

CEMENTS. 
Glue. Powdered chalk added to common glue strengthens it. 
A glue w^hich will resist the action of water is made by boiling 1 
pound of glue in 2 quarts of skimmed milk. 

Soft Cement. For steam-boilers, steam-pipes, &c. Red or white 
lead in oil, 4 parts ; Iron borings, 2 to 3 parts. 

Hard Cement. Iron borings and salt water, and a small quantity 
ot sal ammoniac with fresh water. 



Inside work. Outside work. 

80 
9 



PAINTS. 
White Paint, 

TTTi •ill -. insiae v 

White-lead, ground m oil . . 80. 

Boiled oil 145 

Raw oil Q 

Spirits turpentine .... 8. 4 

New wood work requires about 1 lb. to the square yard for 3 coats. 

Lead Colour. 
White-lead, ground in oil, 75 
Lampblack . . . i 
Boiled linseed oil . 23 



Litharge ... .5 
Japan varnish . . .5 
Spirits turpentine . 2.5 
The turpentine and varnish are added as the paint is required for 
use or transportation. 



White-lead, in oil . 78. 
Boiled oil-. . . 9.5 
Raw oil . . . 9,5 



Gray, or Stone Colour. 



Spirits turpentine . 3. 
Turkey umber . . .5 

Lampblack . . .25 



1 square yard of new brick work requires, for 2 coats, 1.1 lb. • for 
3 coats, 1.5 lbs. * 

Cream Colour. 

White-lead, in oil 
French yellow . 
Japan varnish . 
Raw oil . 
Spirits turpentine 

Uc^^^^oixh^ ^^ ^^^ ^^'""^ ^'^^^ requires, for 1st coat, 0.757 for 



1st coat. 2d coat. 

66.6 70. 

3.3 3.3 

1.3 1.3 

28. 24.5 

225 2.25 



264 MISCELLANEOUS NOTES. 



Lampblack . . 28 

Litharge ... 1 



Black Paint {for Iron). 

Linseed oil, boiled . 73 
Spirits turpentine . 1 



Japan varnish . . 1 

The varnish and turpentine are added last. 





Liquid Olive Colour. 


Olive paste 
Boiled oil . 


. 61.5 
. 29.5 


Dryings 
Japan varnish 


Spirits turpentine 


5.5 





3.5 
2. 



Paint for Tarpaulins {Olive), 
Liquid olive colour . 100 I Spirits turpentine . 6 

Beeswax ... 61 

1 square yard requires 2 lbs. for 3 coats. 
Dissolve the beeswax in the turpentine, and mix the paint warm. 

Lacker for Iron Ordnance. 



Black-lead, pulverized 


12 


Red-lead . 


12 


Litharge . 


5 


Lampblack 


6 


Linseed oil 


66 







Boil it gently for about 20 minutes, stirring it constantly during 
that time. 

Lacker for Small Arms, or for Water Proof Paper. 
Beeswax . . . 18. I Spirits turpentine . 80 
Boiled linseed oil . 3.5 | 

Heat the ingredients in a copper or earthen vessel over a gentle 
fire, in a water bath, until they are well mixed. 

Lacker for Bright Iron Work. 
Linseed oil, boiled . 80.5 I Litharge ... 5.5 
White-leadjgroundinoil, 11.25 I Pulverized rosin . 2.75 
Add the litharge to the oil ; let it simmer over a slow fire for 3 

hours ; strain it, and add the rosin and white-lead ; keep it gently 

warmed, and stir it until the rosici is dissolved 

Staining Wood and Ivor^^ /-^- j A V-A 

Yellow. Dilute nitric acid will produce it on wood. 

Red. An infusion of Brazil wood in stale urine, in the proportion of a lb. to a 
gallon for wood, to be laid on when boiling hot, and should be laid over with alum 
water before it dries. 

Or, a solution of dragon's blood, in spirits of wine, may be used. 

Black. Strong solution of nitric acid, for wood or ivory. 

Mahogany. Brazil, Madder, and Logwood, dissolved in water and put on hot. 

Blue. Ivory may be stained thus : Soak it in a solution of verdigris in nitric 
acid, which will turn it green ; then dip it into a solution of pearlash boiling hot. 

Purple. Soak ivory in a solution of sal ammoniac into four times its weight of 
j[iitrous acid. 



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